On a Schauder basis related to the eigenvectors of a family of non-selfadjoint analytic operators and applications

Abstract

In the present paper, we deal with the perturbed operator

$$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T _1+\varepsilon ^2T_2+\cdots +\varepsilon ^kT_k+\cdots , \end{aligned}$$

where $\varepsilon \in \mathbb C ,\,T_0$ is a closed densely defined linear operator on a separable Banach space $X$ with domain $\mathcal D (T_0),$ while $T_1, T_2, \ldots$ are linear operators on $X$ with the same domain $\mathcal D \supset \mathcal D (T_0)$ and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions assuring the invariance of the closure of the perturbed operator $T(\varepsilon )$ which enables us to study the changed spectrum. Moreover, we prove that the system formed by some eigenvectors of $T(\varepsilon )$ which are analytic on $\varepsilon ,$ forms a Schauder basis in $X.$ After that, we apply the obtained results to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonself-adjoint Gribov operator in Bargmann space.

1 Introduction

Historically, the perturbation method was developed as an approximation device in classical and quantum mechanics. In the perturbation theory of linear operators, the crucial problem is the study of the behavior of spectral properties of linear operators undergo a small change. The foundation of the mathematical theory including a complete convergence proof of perturbation series has been developed by several authors. The most complete results obtained by Kato [11] and Rellich [16] are mainly concerned with the regular perturbation of self-adjoint operators of a Hilbert space, while some attempts have also been made towards the treatment of non-regular cases which are no less important in applications.

Another generalization of the theory was given by Nagy [15]. By his elegant and powerful method of contour integration, he has been able to transfer most of the theorems for self-adjoint operators to a wider class of closed linear operators in Banach space. More precisely, Nagy [15] has considered the perturbed operator

$$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T_1+\varepsilon ^2T_2+\cdots +\varepsilon ^kT_k+\cdots , \end{aligned}$$

(1.1)

where $\varepsilon \in \mathbb C$ and $T_0, T_1, T_2, T_3 \ldots$ are linear operators on $X,$ having the same domain $\mathcal D$ and satisfying the condition

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert +b\Vert T_0\varphi \Vert ) \end{aligned}$$

(1.2)

for all $\varphi \in \mathcal D$ and for all $k\ge 1,$ where $a,b,q > 0.$

Mainly, Nagy has proved in [15], that:

• (i) The series

  $$\begin{aligned} T_0\varphi +\varepsilon T_1 \varphi +\varepsilon ^2 T_2 \varphi +\cdots +\varepsilon ^k T_k\varphi +\cdots \end{aligned}$$

  converges for all $\varphi \in \mathcal D (T_0)$ and for all $|\varepsilon | <q^{-1}.$ Let $T(\varepsilon )\varphi$ be its limit, then $T(\varepsilon )$ is a linear operator with domain $\mathcal D (T_0).$

• (ii) If $T_0$ is closed, then for $\vert \varepsilon \vert <(q+b)^{-1},\,T(\varepsilon )$ is closed.

• (iii) If we denote by $(\lambda _n)_n$ the isolated eigenvalues of the operator $T_0$ with multiplicity one associated to the eigenvectors $(\varphi _n)_n,$ then for each $\lambda _n$ there exists a neighborhood of $\lambda _n$ in which there is a unique eigenvalue $\lambda _n(\varepsilon )$ of the operator $T(\varepsilon )$ with multiplicity one. Moreover, for $\vert \varepsilon \vert$ enough small, $\lambda _n(\varepsilon )$ and the corresponding eigenvector $\varphi _n(\varepsilon )$ of $T(\varepsilon )$ can be developed as entire series of $\varepsilon$ as follows:

  $$\begin{aligned} \lambda _n(\varepsilon )&= \lambda _n +\varepsilon \lambda _n^1+ \varepsilon ^2 \lambda _n^2+\cdots ,\end{aligned}$$

  (1.3)
  $$\begin{aligned} \varphi _n(\varepsilon )&= \varphi _n +\varepsilon \varphi _n^1+\varepsilon ^2 \varphi _n^2+\cdots \end{aligned}$$

  (1.4)

Jeribi and Intissar [9] have exploited the above results due to Nagy to study the existence of a Riesz basis in the Hilbert space $L^2(]-L,L[),$ formed by the eigenvectors of the integro-differential operator

$$\begin{aligned} (I+\varepsilon K)^{-1}\frac{d^4}{dx^4}, \end{aligned}$$

where $L> 0$ and $K$ is the integral operator with kernel the Hankel function of the first kind and order $0.$ In fact, they considered the sequence of operators $(T_k)_{k\ge 0}$ defined for all integer $k\ge 0$ by

$$\begin{aligned} \left\{ \begin{array}{l} T_k:\mathcal{D }(T_k)\subset L^2(]-L,L[)\longrightarrow L^2(]-L,L[)\\ \displaystyle {\varphi \longrightarrow T_k\varphi =\;(-1)^k K^k\frac{d^4\varphi }{dx^4}}\\ \\ \mathcal{D }(T_k)=H_0^2(]-L,L[)\bigcap H^4(]-L,L[). \end{array} \right. \end{aligned}$$

Essentially, they have proved that the system formed by some eigenvectors of the integro-differential operator $(I+\varepsilon _n K)^{-1}\frac{d^4}{dx^4},$ which can be developed as an entire series of $\varepsilon _n,$ forms a Riesz basis in $L^2(]-L,L[).$ Here $(\varepsilon _n)_n$ is a sequence of complex numbers.

Later, in [3] Ben Ali and Jeribi have generalized the above result to the operator $T(\varepsilon )$ [see Eq. (1.1)] in any Hilbert space $X.$ That is, they have proved that the family $(\varphi _n(\varepsilon _n))_n,$ corresponding as a sequence of eigenvectors of the operator $T(\varepsilon _n)$ [see Eq. (1.4)] that can be developed as an entire series of $\varepsilon ,$ forms a Riesz basis in $X.$ As application, they have studied the problem of radiation produced by a vibrating structure in a light fluid [5].

It is along this line of thoughts that we try to formulate some supplements to the results developed in [3, 15] which occur in application to a problem of radiation of a vibrating structure in a light fluid [6] and in Reggeon field theory due to Gribov [7]. More specially, we study the operator $T(\varepsilon )$ [see Eq. (1.1)] where $\varepsilon \in \mathbb C ,\,T_0$ is a closed densely defined linear operator on a separable Banach space $X$ with domain $\mathcal D (T_0),$ while $T_1, T_2, \ldots$ are linear operators on $X$ with the same domain $\mathcal D \supset \mathcal D (T_0)$ and satisfying the following growing inequality

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert +b\Vert T_0\varphi \Vert ^{\beta }\Vert \varphi \Vert ^{1-\beta }) \end{aligned}$$

(1.5)

for all $\varphi \in \mathcal D (T_0)$ and for all $k\ge 1,$ with $\beta \in ]0,1]$ and $a,b,q > 0.$ This new condition enables us to give a considerable improvement to the estimation of the convergence radius for $\lambda _n(\varepsilon )$ and $\varphi _n(\varepsilon )$ and to the estimation of the coefficients $(\lambda _n^i)_{i\ge 1}$ and $(\varphi _n^i)_{i\ge 1}$ [see Eqs. (1.3), (1.4)]. Using the obtained estimations, we investigate under sufficient conditions assuring the completeness of the system of eigenvectors of the operator $T(\varepsilon )$ (see Theorem 4.2). Furthermore, we extend the main result in [3] to Schauder basis in any Banach space. In this context we prove that there exist a sequence of complex $(\varepsilon _n)_n$ and a sequence $(\varphi _n(\varepsilon ))_n$ of eigenvectors of $T(\varepsilon )$ as expressed in Eq. (1.4), such that the system $(\varphi _n(\varepsilon _n))_n$ forms a Schauder basis in $X.$

The obtained results are of importance for application to a problem of sound radiation produced by a vibrating plate in a light fluid [6]. Indeed, we consider the following eigenvalue problem: Find the values $\lambda (\varepsilon )\in \mathbb C$ for which there is a solution $\varphi \in H_0^2(]-L,L[)\cap H^4(]-L,L[),\,\varphi \ne 0,$ for the equation

$$\begin{aligned} \left( \frac{d^4}{dx^4}-\varepsilon K\frac{d^2}{dx^2}+ \varepsilon ^2 K^2\frac{d^2}{dx^2}+\cdots +(-1)^n\varepsilon ^n K^n\frac{d^2}{dx^2}+\cdots \right) \varphi =\lambda (\varepsilon )\varphi , \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{l} K:L^2(]-L,L[)\longrightarrow L^2(]-L,L[)\\ \displaystyle {\varphi \longrightarrow K\varphi (x)=\frac{i}{2}\int _{-L}^L H_0(k\vert x-x^{\prime }\vert )\varphi (x^{\prime })dx^{\prime }}, \end{array} \right. \end{aligned}$$

$L> 0$ and $H_0$ is the Hankel function of the first kind and order $0.$ Setting $(\varepsilon _n)_n$ a sequence of complex numbers, we prove that for $\vert \varepsilon _n\vert$ enough small, the system of eigenvectors of the family of operators

$$\begin{aligned} \frac{d^4}{dx^4}-\varepsilon _n K\frac{d^2}{dx^2}+ \varepsilon _n^2 K^2\frac{d^2}{dx^2}+\cdots +(-1)^k\varepsilon _n^k K^k\frac{d^2}{dx^2}+\cdots \end{aligned}$$

forms a Schauder basis in the Hilbert space $L^2(]-L,L[),$ which gives a positive answer to the hypotheses considered in [6].

Moreover, an application to a Gribov operator in Bargmann space (see [1, 2, 10]) introduced by Gribov in his Reggeon field theory [7] is considered in this paper. In that line, we assure the existence of a Schauder basis of the Bargmann space

$$\begin{aligned} E=\left\{ \varphi :\mathbb C \longrightarrow \mathbb C \text{ entire } \text{ such } \text{ that } \displaystyle \int _\mathbb{C }e^{-\vert z\vert ^2}\vert \varphi (z)\vert ^2 dzd\bar{z}<\infty \quad \text{ and }\quad \varphi (0)=0\right\} \end{aligned}$$

formed by eigenvectors of the family of the following Gribov operator

$$\begin{aligned} (A^{*}A)^3+\varepsilon _n A^{*}(A+A^{*})A+\varepsilon _n^2(A^{*}A)^{3u_2}+\cdots +\varepsilon _n^k(A^{*}A)^{3u_k}+\cdots , \end{aligned}$$

where $A$ (respectively, $A^{*}$) is the annihilator (respectively, creator) operator, $\varepsilon \in \mathbb C$ and $(u_k)_{k\in \mathbb N }$ is a strictly decreasing sequence with strictly positive terms such that $u_0=1$ and $u_1=\frac{1}{2}.$

We now describe briefly the contents of this paper section by section. In Sect. 2, we prove that if $T_0$ is closed, then it is so for the perturbed operator $T(\varepsilon ),$ for $\vert \varepsilon \vert$ enough small. Section 3 is devoted to the study of the invariance of the spectrum. Indeed, we prove that for each isolated eigenvalue $\lambda _n$ of $T_0$ with multiplicity one, there exists a neighborhood of $\lambda _n$ in which there is a unique eigenvalue $\lambda _n(\varepsilon )$ of the operator $T(\varepsilon )$ with multiplicity one. Besides, if $\varphi _n(\varepsilon )$ is an eigenvector of $T(\varepsilon )$ associated to $\lambda _n(\varepsilon ),$ then $\lambda _n(\varepsilon )$ and $\varphi _n(\varepsilon )$ can be developed as entire series of $\varepsilon .$ In Sect. 4, we establish the completeness of the system of root vectors of the operator $T(\varepsilon ).$ In Sect. 5, we prove that the system formed by some eigenvectors of the operator $T(\varepsilon _n),$ that can be developed as an entire series of $\varepsilon _n,$ forms a Schauder basis of $X.$ Finally, in Sect. 6, we apply the obtained abstract results to a problem of radiation of a vibrating structure in a light fluid (see [3, 4, 9]) and to a problem arising from reggeon field theory motivated by Gribov [7].

2 Invariance of the closure

Let $T_0$ be a linear operator on a Banach space $X$ such that: $(H1)\,T_0$ is closed with domain $\mathcal D (T_0)$ dense in $X.$ Let $T_1, T_2, T_3, \ldots$ be some linear operators on $X$ having the same domain $\mathcal D$ and satisfying the hypothesis:

$(H2)\,\mathcal D (T_0) \subset \mathcal D$ and there exist $a, b, q> 0$ and $\beta \in ]0,1]$ such that for all $k\ge 1$

$$\begin{aligned} \Vert T_k \varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert + b\Vert T_0 \varphi \Vert ^{\beta } \Vert \varphi \Vert ^{1-\beta }), \;\; \text{ for } \text{ all } \; \varphi \in \mathcal D (T_0). \end{aligned}$$

(2.1)

The main result of this section is formulated as follow:

Theorem 2.1

Assume that the assumptions (H1) and (H2) hold. Then for $\vert \varepsilon \vert <q^{-1},$ the series

$$\begin{aligned} \sum _{k\ge 0}\varepsilon ^kT_k \varphi \end{aligned}$$

converges for all $\varphi \in \mathcal D (T_0).$ If $T(\varepsilon )\varphi$ denotes its limit, then $T(\varepsilon )$ is a linear operator with domain $\mathcal D (T_0)$ and for $\vert \varepsilon \vert <(q+\beta b)^{-1},$ the operator $T(\varepsilon )$ is closed. $\square$

To prove Theorem 2.1, we need the following lemma:

Lemma 2.1

Let $A$ and $B$ be two linear operators from a Banach space $X$ into a Banach space $Y,$ having the same domain $\mathcal D \subset X.$ Let $A$ be a closed operator. If there exist positive constants $\theta _1, \theta _2$ and $\beta$ satisfying $\beta \in ]0,1]$ and $\theta _2<\frac{1}{\beta }$ such that

$$\begin{aligned} \Vert (A-B)\varphi \Vert \le \theta _1 \Vert \varphi \Vert +\theta _2 \Vert A\varphi \Vert ^{\beta } \Vert \varphi \Vert ^{1-\beta }\qquad {for\, all}\,\varphi \in \mathcal D , \end{aligned}$$

(2.2)

then $B$ is also closed. $\square$

Proof of Lemma 2.1

Let $\varphi \in \mathcal D .$ Eq. (2.2) implies that

$$\begin{aligned} \Vert B \varphi \Vert&\le \Vert A\varphi \Vert + \Vert (B-A)\varphi \Vert \nonumber \\&\le \Vert A\varphi \Vert + \theta _1 \Vert \varphi \Vert + \theta _2 \Vert A\varphi \Vert ^\beta \Vert \varphi \Vert ^{1-\beta }, \end{aligned}$$

(2.3)

and that

$$\begin{aligned} \Vert A \varphi \Vert&\le \Vert B\varphi \Vert + \Vert (A-B)\varphi \Vert \nonumber \\&\le \Vert B\varphi \Vert + \theta _1 \Vert \varphi \Vert + \theta _2 \Vert A\varphi \Vert ^\beta \Vert \varphi \Vert ^{1-\beta }. \end{aligned}$$

(2.4)

Using Young’s inequality, we have

$$\begin{aligned} \Vert A\varphi \Vert ^\beta \Vert \varphi \Vert ^{1-\beta }\le \beta \Vert A\varphi \Vert +(1-\beta )\Vert \varphi \Vert . \end{aligned}$$

(2.5)

So, substituting Eq. (2.5) in Eqs. (2.3) and (2.4), we get

$$\begin{aligned} \Vert B\varphi \Vert \le (1+\theta _2 \beta )\Vert A\varphi \Vert + (\theta _1+\theta _2(1-\beta ))\Vert \varphi \Vert , \end{aligned}$$

(2.6)

and

$$\begin{aligned} (1-\theta _2\beta )\Vert A\varphi \Vert \le \Vert B\varphi \Vert +(\theta _1+\theta _2(1-\beta ))\Vert \varphi \Vert . \end{aligned}$$

(2.7)

Since $\theta _2<{\frac{1}{\beta }},$ then $1-\theta _2\beta >0,$ so Eq. (2.7) implies that

$$\begin{aligned} \Vert A\varphi \Vert \le \frac{1}{1-\theta _2\beta }\Vert B\varphi \Vert +\frac{\theta _1+\theta _2(1-\beta )}{1-\theta _2\beta }\Vert \varphi \Vert . \end{aligned}$$

(2.8)

Now, let $(\varphi _n)_n$ be a sequence in $\mathcal D$ such that $\varphi _n\rightarrow \varphi$ as $n\rightarrow \infty$ and $B\varphi _n \rightarrow g$ as $n\rightarrow \infty ;$ and let us prove that $\varphi \in \mathcal D$ and $g=B\varphi .$

Using Eq. (2.8), we have

$$\begin{aligned} \Vert A(\varphi _n-\varphi _m)\Vert&\le \displaystyle \frac{1}{1-\theta _2\beta }\Vert B(\varphi _n-\varphi _m)\Vert +\frac{\theta _1+\theta _2(1-\beta )}{1-\theta _2\beta }\\&\times \Vert \varphi _n-\varphi _m\Vert \rightarrow 0 \text{ as } m,n \rightarrow \infty . \end{aligned}$$

So, the sequence $(A\varphi _n)_n$ is a Cauchy sequence in $Y.$ Hence, $(A\varphi _n)_n$ is convergent. On the other hand, $A$ is closed, so, $\varphi \in \mathcal D$ and $A\varphi _n\rightarrow A\varphi$ as $n\rightarrow \infty .$ Using Eq. (2.6), we have

$$\begin{aligned} \Vert B(\varphi _n\!-v\varphi )\Vert \le (1+\theta _2 \beta )\Vert A (\varphi _n\!-\!\varphi )\Vert \!+\! (\theta _1\!+\!\theta _2(1-\beta ))\Vert \varphi _n\!-\!\varphi \Vert \!\rightarrow \! 0 \text{ as } n\!\rightarrow \!\infty . \end{aligned}$$

So, $B\varphi _n\rightarrow B\varphi \text{ as } n\rightarrow \infty$ and $g=B\varphi .$ Consequently, $B$ is closed. $\square$

Proof of Theorem 2.1

Let $\vert \varepsilon \vert < \frac{1}{q}$  and $\varphi \in \mathcal D .$ It follows from Eq. (2.1) that

$$\begin{aligned} \left\| {\sum _{k=0}^{\infty }} \varepsilon ^k T_k\varphi \right\|&\le \displaystyle {\sum _{k=0}^{\infty }} \Vert \varepsilon ^k T_k\varphi \Vert \\&\le \displaystyle {\sum _{k=0}^{\infty }} \vert \varepsilon \vert ^k \Vert T_k\varphi \Vert \\&\le \Vert T_0\varphi \Vert \ + \displaystyle {\sum _{k=1}^{\infty }} \vert \varepsilon \vert ^k \Vert T_k\varphi \Vert \\&\le \Vert T_0\varphi \Vert + {\sum _{k=1}^{\infty }} \vert \varepsilon \vert ^k q^{k-1}(a \Vert \varphi \Vert +b \Vert T_0\varphi \Vert ^\beta \Vert \varphi \Vert ^{1-\beta }) \\&\le \Vert T_0\varphi \Vert \ + {\frac{\vert \varepsilon \vert }{1-\vert \varepsilon \vert q}}(a \Vert \varphi \Vert +b \Vert T_0\varphi \Vert ^\beta \Vert \varphi \Vert ^{1-\beta }). \end{aligned}$$

So, the series $\sum _{k\ge 0}\varepsilon ^k T_k\varphi$ is convergent. Similarly, using Eq. (2.1), we have

$$\begin{aligned} \Vert (T(\varepsilon )-T_0)\varphi \Vert \le \theta _1(\varepsilon ) \Vert \varphi \Vert +\theta _2(\varepsilon ) \Vert T_0\varphi \Vert ^{\beta } \Vert \varphi \Vert ^{1-\beta }, \text{ for } \text{ all } \varphi \in \mathcal D (T_0), \end{aligned}$$

where $\theta _1(\varepsilon ) =\frac{a\vert \varepsilon \vert }{1-\vert \varepsilon \vert q}$   and    $\theta _2(\varepsilon )=\frac{b\vert \varepsilon \vert }{1-\vert \varepsilon \vert q}.$

If $\vert \varepsilon \vert <{\frac{1}{q+\beta b}},$ we have $\theta _2(\varepsilon ) < {\frac{1}{\beta }}.$ Since $T_0$ is a closed operator, we deduce, in view of Lemma 2.1, that the operator $T(\varepsilon )$ is closed for $\vert \varepsilon \vert <{\frac{1}{q+\beta b}}.$ $\square$

Remark 2.1

Theorem 2.1 $($respectively, Lemma 2.1$)$ may be regarded as an extension of [15, Théorème 3] $($respectively, [15, Théorème 1] $).$ $\square$

3 Perturbation of the spectrum

Let $n\in \mathbb N ^{*},\,\lambda _n$ the isolated eigenvalue number $n$ of the operator $T_0$ with multiplicity one, $d_n=d(\lambda _n, \sigma (T_0){\setminus }\{\lambda _n\})\!:$ the distance between $\lambda _n$ and $\sigma (T_0){\setminus }\{\lambda _n\}$ and $\mathcal C _n=\mathcal C (\lambda _n, r_n)\!:$ the closed circle with center $\lambda _n$ and with radii $r_n=\frac{d_n}{2}.$ Since $(T_0-z I)^{-1}$ is a regular analytic function of $z,\,\Vert (T_0-z I)^{-1} \Vert$ is a continuous function of $z.$ So, we denote by:

$$\begin{aligned} M_n:=\max _{z\in \mathcal C _n}\Vert (T_0-z I)^{-1}\Vert \end{aligned}$$

and

$$\begin{aligned} N_n:= \max _{z\in \mathcal C _n}\Vert T_0(T_0-z I)^{-1}\Vert = \max _{z\in \mathcal C _n} \Vert I+z(T_0-z I)^{-1}\Vert . \end{aligned}$$

Theorem 3.1

Assume that the hypotheses (H1) and (H2) hold. Let $\varphi _n$ $($respectively, $\varphi _n^{*}$ $)$ be an eigenvector of $T_0$ $($respectively, $T_0^{*}\!:$ the adjoint of $T_0\,)$ associated to the eigenvalue $\lambda _n$ $($respectively, $\overline{\lambda _n}$ $)$ such that $\Vert \varphi _n\Vert =\Vert \varphi _n^{*}\Vert$ and $\varphi _n^{*}(\varphi _n)=1.$ Then:

• (i) For $\vert \varepsilon \vert <\frac{1}{q+\alpha _n+r_nM_n\alpha _n},\,T(\varepsilon )$ has a unique point of the spectrum in the neighborhood of $\lambda _n,$ and this point $\lambda _n (\varepsilon )$ is also with multiplicity one.

• (ii) For $\vert \varepsilon \vert <\frac{1}{q+\alpha _n+\omega _n^2r_nM_n\alpha _n},\,\lambda _n(\varepsilon )$ and the corresponding eigenvector $\varphi _n(\varepsilon )$ of $T(\varepsilon )$ can be developed into an entire series of $\varepsilon \!:$

  $$\begin{aligned} \lambda _n(\varepsilon )&= \lambda _n +\varepsilon \lambda _n^1+ \varepsilon ^2 \lambda _n^2+\cdots ,\end{aligned}$$

  (3.1)
  $$\begin{aligned} \varphi _n(\varepsilon )&= \varphi _n +\varepsilon \varphi _n^1+\varepsilon ^2 \varphi _n^2+\cdots , \end{aligned}$$

  (3.2)

and we have

$$\begin{aligned}&\vert \lambda _n^i\vert \le \omega _n^2 r_n^2 M_n \alpha _n (q+\alpha _n+\omega _n^2 r_nM_n\alpha _n)^{i-1}\quad {for\,all} i\ge 1,\end{aligned}$$

(3.3)

$$\begin{aligned}&\Vert \varphi _n^i\Vert \le \omega _n r_n M_n(q+\alpha _n +\omega _n^2 r_nM_n\alpha _n)^{i}\quad {for\, all} i\ge 1, \end{aligned}$$

(3.4)

where $\omega _n=\Vert \varphi _n\Vert$ and $\alpha _n:=aM_n+bN_n^{\beta }M_n^{1-\beta }.$ $\square$

Remark 3.1

Nagy [15] has studied the operator $T(\varepsilon )$ with the coefficients $(T_k)_{k\ge 1}$ satisfy Eq. (2.1) with $\beta =1.$ If in particular we take $\beta =1$ in Theorem 3.1, we regain the results of [15, Partie 2]. The advantage of considering the estimation (2.1) with $\beta \in ]0,1]$ instead of $\beta =1$ is the improvement of the convergence radius for $\lambda _n(\varepsilon )$ and $\varphi _n(\varepsilon ).$ $\square$

Proof

Let $z_n\in \mathcal C _n.$ Using Eq. (2.1), we infer that for all $g\in X$ and for all $k\ge 1,$

$$\begin{aligned} \Vert T_k(T_0-z_n I)^{-1}g\Vert&\le q^{k-1}(a\Vert (T_0-z_n I)^{-1}g\Vert + b\Vert T_0(T_0-z_n I)^{-1}g\Vert ^\beta \\&\Vert (T_0-z_n I)^{-1}g\Vert ^{1-\beta })\\&\le q^{k-1}(a\Vert (T_0-z_n I)^{-1}\Vert + b\Vert T_0(T_0-z_n I)^{-1}\Vert ^\beta \\&\Vert (T_0-z_n I)^{-1}\Vert ^{1-\beta })\Vert g\Vert . \end{aligned}$$

So,

$$\begin{aligned} \Vert T_k(T_0-z_n I)^{-1}\Vert \le q^{k-1}\alpha _n. \end{aligned}$$

We can easily see that for $\vert \varepsilon \vert <{\frac{1}{q+\alpha _n}},\, \rho (T_0)\subset \rho (T(\varepsilon )),$ in particular we have

$$\begin{aligned} \mathcal C _n\subset \rho (T(\varepsilon )). \end{aligned}$$

Now, the rest of the proof may be sketched in a similar way to that in [15]. It suffices to replace $\alpha :=aM+bN$ (see [15, p. 133]) by $\alpha _n:=aM_n+bN_n^{\beta }M_n^{1-\beta }.$ $\square$

4 Completeness of the system of root vectors of $T(\varepsilon )$

In this section, we prove that the system of root vectors of the operator $T(\varepsilon )$ is complete in $X.$ We first recall some definitions and preliminary results.

Definition 4.1

Let $A$ be a linear compact operator on a Banach space $X.$ The $n^{th}$ approximation number $a_n(A)$ of $A$ is defined as:

$$\begin{aligned} a_n(A):=\inf \,\left\{ \Vert A-T\Vert \text{ such } \text{ that } rk(T)\le n \right\} \!, \end{aligned}$$

where $rk(T)$ designates the rank of $T.$ $\square$

Remark 4.1

We shall emphasize on the fact that if $X$ is a Hilbert space, then the approximation numbers ($a$-numbers) of a linear compact operator $A$ coincide with the singular numbers ($s$-numbers)(see [8, p. 26]) of $A$ (see [8, Theorem 2.1, p. 28] or [13, p. 216]).$\square$

Definition 4.2

[13, p. 168] Let $A$ be a linear compact operator on a Banach space $X.\,A$ is said to belong to the class $\mathcal C _p(X),\,0<p<\infty ,$ if the series $\sum _{n= 1}^{\infty }(a_n(A))^p$ converges, where $a_n(A), \, n=1,2,\ldots$ are the approximation numbers of $A.$ $\square$

Definition 4.3

[13, p. 212] Let $A$ be a linear operator on a Banach space $X.\,A$ is called quasiconservative if its spectrum is purely imaginary and there exists a constant $M\ge 0$ such that

$$\begin{aligned} \Vert (A-\lambda I)^{-1}\Vert \le \frac{M}{\vert Re \lambda \vert },~ Re \lambda \ne 0, \text{ for } \text{ all } \lambda \in \rho (A). \end{aligned}$$

$\square$

Definition 4.4

Let $A$ be a linear operator on a Banach space $X.\,A$ is said to have quasiconservative resolvent if the resolvent set of $A,\,\rho (A),$ is not empty and there exists $\lambda _0\in \rho (A)$ such that $(A-\lambda _0 I)^{-1}$ is quasiconservative.$\square$

Theorem 4.1

[13, Theorem p. 219] Let $A$ be a quasiconservative linear operator on a Banach space $X.$ We suppose that the operator $A$ belongs to the class $\mathcal{C }_{p}(X),\,0<p<\infty .$ Let $B=A(I+K),$ where $K$ is a compact operator in $X.$ If the operator $B$ vanishes only at zero, then the system of its root vectors is complete in $X.$ $\square$

Definition 4.5

[17, Definition 2.2, p. 50] Let $A$ and $B$ be two linear operators on a Banach space $X$ such that $A$ is closed. The operator $B$ is said to be $A$-compact if $\mathcal D (A)\subset \mathcal D (B)$ and for all $\varepsilon >0,$ there exists a constant $C(\varepsilon )>0$ such that

$$\begin{aligned} \Vert B\varphi \Vert \le \varepsilon \Vert A\varphi \Vert + C(\varepsilon )\Vert \varphi \Vert \quad \text{ for } \text{ all } \varphi \in \mathcal D (A). \end{aligned}$$

$\square$

Proposition 4.1

[17, Lemma 2.5, p. 52] Let $A$ and $B$ be two linear operators on a Banach space $X$ such that $A$ is closed with nonempty resolvent set. If $B$ is $A$-compact and $A$ is with compact resolvent, then $B(A-\lambda I)^{-1}$ is compact, where $\lambda \in \rho (A).$ $\square$

Now, we are ready to state the first result of this section:

Theorem 4.2

Assume that the hypotheses (H1) and (H2) hold. In addition, suppose that the resolvent of $T_0$ is quasiconservative and belongs to the class $\mathcal C _p(X), 0<p<\infty .$ If $T_i$ is $T_0$-compact for all $i\in \mathbb N ^{*},$ then, for $\vert \varepsilon \vert$ enough small, the system of root vectors of the operator $T(\varepsilon )$ is complete in $X.$ $\square$

Remark 4.2

Theorem 4.2 generalizes Proposition $3.1$ in [3] to any Banach space.$\square$

Proof of Theorem 4.2

Since the resolvent of $T_0$ is quasiconservative, there exists $\lambda _0\in \rho (T_0)$ such that $(T_0-\lambda _0 I)^{-1}$ is quasiconservative. We have

$$\begin{aligned} T(\varepsilon )-\lambda _0 I&= T_0-\lambda _0 I +\varepsilon T_1+ \varepsilon ^2 T_2+\cdots \nonumber \\&= (I+\varepsilon T_1(T_0-\lambda _0I)^{-1}+ \varepsilon ^2 T_2(T_0-\lambda _0I)^{-1}+\cdots )(T_0-\lambda _0I) \nonumber \\&= (I+S)(T_0-\lambda _0I), \end{aligned}$$

(4.1)

where

$$\begin{aligned} S:= \sum _{k=1}^{\infty }\varepsilon ^k T_k(T_0-\lambda _0I)^{-1}. \end{aligned}$$

Using Eq. (1.5), we get

$$\begin{aligned} \Vert S\Vert&\le \sum _{k=1}^{\infty }\vert \varepsilon \vert ^k \Vert T_k(T_0-\lambda _0I)^{-1}\Vert \\&\le M(\lambda _0, \beta , a, b)\sum _{k=1}^{\infty }\vert \varepsilon \vert ^k q^{k-1}, \end{aligned}$$

where $M(\lambda _0, \beta ,a, b)=a\Vert (T_0-\lambda _0I)^{-1}\Vert +b \Vert I+\lambda _0(T_0-\lambda _0I)^{-1}\Vert ^{\beta } \Vert (T_0-\lambda _0I)^{-1}\Vert ^{1-\beta }.$

For $\vert \varepsilon \vert < q^{-1},$ we have

$$\begin{aligned} \Vert S\Vert \le \frac{M(\lambda _0,\beta , a, b)\vert \varepsilon \vert }{1-\vert \varepsilon \vert q} \end{aligned}$$

and therefore for $\vert \varepsilon \vert <\frac{1}{q+M(\lambda _0, \beta , a, b)},$ we have $\Vert S\Vert <1$ and $I+S$ is invertible. So Eq. (4.1), implies that $T(\varepsilon )-\lambda _0I$ is invertible and

$$\begin{aligned} (T(\varepsilon )-\lambda _0 I )^{-1}&= (T_0-\lambda _0I)^{-1} (I+ S)^{-1} \\&= (T_0-\lambda _0I)^{-1}(I+K), \end{aligned}$$

where $K=-S+S^2-S^3+...~.$ Since the resolvent of $T_0$ is compact and $T_k$ is $T_0$-compact for all $k\ge 1,$ then due to Proposition 4.1, $T_k(T_0-\lambda _0I)^{-1}$ is compact for all $k\ge 1.$ So, the operator $K$ is compact on $X.$ Consequently, the result follows from Theorem 4.1. $\square$

Similarly, if $X$ is a Hilbert space, then making the same reasoning as the one in the proof of Theorem 4.2 and using Keldys’ Theorem (see [13, Theorem 8.1, p. 257]), we have the following result:

Theorem 4.3

Assume that the hypotheses (H1) and (H2) hold. In addition, suppose that $T_0$ is self-adjoint and its resolvent belongs to the class $\mathcal C _p(X),\,0<p<\infty .$ If $T_i$ is $T_0$-compact for all $i\in \mathbb N ^{*},$ then, for $\vert \varepsilon \vert$ enough small, the system of root vectors of the operator $T(\varepsilon )$ is complete in $X.$ $\square$

5 On a Schauder basis in a separable Banach space $X$

The aim of this section, is to prove that the system of eigenvectors of the perturbed operator $T(\varepsilon )$ forms a Schauder basis in a separable Banach space $X.$ To this interest, we give the following definitions and preliminary results:

Definition 5.1

[18, Definition  p. 1] A sequence of vectors $\{\varphi _n\}_{n\in \mathbb N ^{*}}$ in an infinite-dimensional Banach space $X$ is said to be a Schauder basis or a basis for $X$ if to each vector $x\in X$ there corresponds a unique sequence of scalars $\{c_n\}_{n\in \mathbb N ^{*}}$ such that

$$\begin{aligned} x=\sum _{n=1}^{\infty }c_n\varphi _n. \end{aligned}$$

$\square$

Remark 5.1

Each coefficient $c_n$ is a linear function of $x.$ If we denote this linear function by $f_n,$ then $c_n=f_n(x),$ and we may write

$$\begin{aligned} x=\sum _{n=1}^{\infty }f_n(x)\varphi _n. \end{aligned}$$

The functionals $\{f_n\}_{n\in \mathbb N ^{*}}$ thus defined, are called the coefficient functionals associated with the basis $\{\varphi _n\}_{n\in \mathbb N ^{*}}.$ $\square$

Theorem 5.1

(see [12] or [18, Theorem 11 p. 39]) If $\{\varphi _n\}_{n\in \mathbb N ^{*}}$ is a basis for a Banach space $X$ and $\{f_n\}_{n\in \mathbb N ^{*}}$ is the sequence of coefficient functionals associated with the basis $\{\varphi _n\}_{n\in \mathbb N ^{*}},$ then each system of vectors $\{\psi _n\}_{n\in \mathbb N ^{*}}$ satisfying the condition

$$\begin{aligned} \sum _{n=1}^\infty \Vert f_n\Vert \Vert \varphi _n-\psi _n\Vert <1, \end{aligned}$$

is a basis for $X.$ $\square$

The objective of this section is formulated as:

Theorem 5.2

Assume that the hypotheses (H1) and (H2) hold, the eigenvalues $(\lambda _n)_n$ of $T_0$ are isolated with multiplicity one and the corresponding eigenvectors $(\varphi _n)_n$ of $T_0$ form a basis in $X,$ then there exist a sequence of complex $(\varepsilon _n)_{n\in \mathbb N ^*}$ and a sequence $\{\varphi _n(\varepsilon _n)\}_{n\in \mathbb N ^{*}}$ having the form

$$\begin{aligned} \varphi _n(\varepsilon _n)=\varphi _n+\varepsilon _n \varphi _n^1+\varepsilon _n^2 \varphi _n^2+\cdots \end{aligned}$$

such that the system $\{\varphi _n(\varepsilon _n)\}_{n\in \mathbb N ^*}$ forms a basis in $X.$ $\square$

Remark 5.2

Theorem 5.2 extends the main result in [3] to any Banach space. $\square$

Proof of Theorem 5.2

Let $n\in \mathbb N ^{*}$ and $\lambda _n$ the eigenvalue number $n$ of the operator $T_0.$ Due to Theorem 3.1, for

$$\begin{aligned} \vert \varepsilon \vert <{\frac{1}{q+\alpha _n+\omega _n^2r_nM_n\alpha _n}}, \end{aligned}$$

the perturbed transformation $T(\varepsilon )$ will have a unique point of the spectrum inside of the circle $\mathcal C _n$ and this point $\lambda _n(\varepsilon )$ will be also with multiplicity one. The eigenvalue $\lambda _n(\varepsilon )$ and the corresponding eigenvector $\varphi _n(\varepsilon )$ of $T(\varepsilon )$ can be developed as entire series of $\varepsilon$ [Eqs. (3.1) and (3.2)] and we have the estimations (3.3) and (3.4). On the other hand, since the eigenvectors $(\varphi _n)_n$ of $T_0$ form a Schauder basis for $X;$ so, we denote by $(\psi _n)_n$ the coefficient functionals associated with the basis $\{\varphi _n\}_{n\in \mathbb N ^{*}}$ and $\theta _n>0$ such that

$$\begin{aligned} \sum _{n=1}^{\infty }\theta _n\Vert \psi _n\Vert <1. \end{aligned}$$

For each eigenvalue $\lambda _n$ of $T_0,$ we fix an $\varepsilon _n \in \mathbb C$ such that

$$\begin{aligned} \vert \varepsilon _n\vert \in \left]0, {\frac{\theta _n}{(\omega _n r_nM_n+\theta _n)(q+\alpha _n +\omega _n^2r_nM_n\alpha _n)}}\right[. \end{aligned}$$

Since

$$\begin{aligned} \left]0,\frac{\theta _n}{(\omega _n r_nM_n+\theta _n)(q+\alpha _n +\omega _n^2r_nM_n\alpha _n)}\right[\subset \left]0,{\frac{1}{q+\alpha _n +\omega _n^2r_nM_n\alpha _n}}\right[, \end{aligned}$$

then, the eigenvalue $\lambda _n(\varepsilon _n)$ and the eigenvector $\varphi _n(\varepsilon _n)$ can be developed as entire series of $\varepsilon _n\!:$

$$\begin{aligned} \lambda _n(\varepsilon _n)&= \lambda _n +\varepsilon _n \lambda _n^1+ \varepsilon _n^2 \lambda _n^2+\cdots ,\\ \varphi _n(\varepsilon _n)&= \varphi _n +\varepsilon _n \varphi _n^1+\varepsilon _n^2 \varphi _n^2+\cdots . \end{aligned}$$

Using the above equation and Eq. (3.4), we obtain

$$\begin{aligned} \Vert \varphi _n(\varepsilon _n)-\varphi _n\Vert&\le {\sum _{i=1}^{\infty }}\vert \varepsilon _n\vert ^i\omega _n r_n M_n(q+\alpha _n+\omega _n^2r_nM_n\alpha _n)^i\\&\le \omega _n r_n M_n{\sum _{i=1}^{\infty }}(\vert \varepsilon _n\vert (q+\alpha _n+\omega _n^2r_nM_n\alpha _n))^i\\&\le {\omega _n r_n M_n\frac{\vert \varepsilon _n\vert (q+\alpha _n+\omega _n^2r_nM_n\alpha _n)}{1-\vert \varepsilon _n\vert (q+\alpha _n+\omega _n^2r_nM_n\alpha _n)}}\\&< \theta _n. \end{aligned}$$

Using Krein–Milman–Rutman’s theorem (see Theorem 5.1), we deduce that the system $\{\varphi _n(\varepsilon _n)\}_{n\in \mathbb N ^*}$ forms a basis of the space $X.$ $\square$

6 Applications

6.1 Application to a problem of radiation of a vibrating structure in a light fluid

We consider an elastic membrane lying in the domain $-L<x<L$ of the plane $z=0.$ The two half-spaces $z<0$ and $z>0$ are filled with gas. The membrane is excited by a harmonic force $F(x)e^{-i\omega t}$ and its mechanical parameters are:

• $E:$ Young modulus.

• $\nu :$ Poisson ratio.

• $m:$ surface density.

• $h:$ thickness of the membrane.

• $D:\,=\frac{Eh^3}{12(1-\nu ^2)}:$ the rigidity. The fluid is characterized by:

• $\rho _0:$ density.

• $c:$ sound speed.

• $k:\,=\frac{\omega }{c}:$ the wave number.

Let $u$ denote the displacement of the membrane and $p$ the acoustic pressure in the fluid.

These functions satisfy the system:

$$\begin{aligned} \left( \frac{d^4}{dx^4}-\frac{m\omega ^2}{D}\right) u(x)=\frac{1}{D}\left( F(x)-P(x)\right) \text{ for } \text{ all } \varphi \in ]-L,L[, \end{aligned}$$

(6.1)

where

$$\begin{aligned}&u(x) = \frac{\partial u(x)}{\partial x} = 0 \text{ for } x=-L \text{ and } x=L,\end{aligned}$$

(6.2)

$$\begin{aligned}&P(x)=\lim \limits _{z\rightarrow 0^+}\left( p(x,z)-p(x,-z)\right) \end{aligned}$$

(6.3)

and

$$\begin{aligned} p(x,z)&= -sgn zi\frac{\rho _0}{2}\int _{-L}^{L} H_0\left( k \sqrt{(x-x^{\prime })^2 + z^2}\right) \nonumber \\&\times \left( \omega ^2- \frac{D}{m}\left( \frac{d^4}{dx^4}-\frac{d^2}{dx^2}\right) \right) u(x^{\prime })\mathrm d x^{\prime }, \end{aligned}$$

(6.4)

for $z<0$ or $z>0,$ with $H_0$ designates the Hankel function of the first kind and order $0.$

Expressions (6.1), (6.2), (6.3) and (6.4) lead to the boundary value problem

$$\begin{aligned}&\left( \frac{d^4}{dx^4}-\frac{m\omega ^2}{D}\right) u(x) -i\rho _0\int _{-L}^{L} H_0\left( k\vert x-x^{\prime } \vert \right) \left( \frac{\omega ^2}{D}-\frac{1}{m}\left( \frac{d^4}{dx^4}-\frac{d^2}{dx^2} \right) \right) \nonumber \\&\quad \times \, u(x^{\prime }) dx^{\prime }=\frac{F(x)}{D}, \end{aligned}$$

(6.5)

for all $x\in ]-L,L[$ with $u(x) = \frac{\partial u(x)}{\partial x} = 0\,\, \text{ for } \,\, x=-L \,\, \text{ and } \,\, x=L.$

Let the operator

$$\begin{aligned} \left\{ \begin{array}{l} T_0:\mathcal{D }(T_0)\subset L^2(]-L,L[)\longrightarrow L^2(]-L,L[)\\ {\varphi \longrightarrow T_0\varphi (x)=\frac{d^4\varphi }{dx^4}}\\ \\ \mathcal{D }(T_0)=H_0^2(]-L,L[)\cap H^4(]-L,L[). \end{array} \right. \end{aligned}$$

Remark 6.1

Due to [9, Lemma   3.2], $T_0$ is a self-adjoint operator and has a compact resolvent. Then, let

$$\begin{aligned} T_0\varphi = \sum _{n=1}^\infty \lambda _n\langle \varphi ,\varphi _n\rangle \varphi _n \end{aligned}$$

be its spectral decomposition, where $\lambda _n=\alpha n^4$ is the $n^\mathrm{th}$ eigenvalue of $T_0$ associated to the eigenvector $\varphi _n(x)=\mu e^{\root 4 \of {\lambda _n}x}+\eta e^{-\root 4 \of {\lambda _n}x}+\theta e^{i\root 4 \of {\lambda _n}x}+\delta e^{\root 4 \of {\lambda _n}x}$ (see [9, p. 7]).

$\square$

Hence, for $\gamma >0,$ we define the operator $T_0^\gamma$ by

$$\begin{aligned} \left\{ \begin{array}{l} T_0^\gamma :{ \mathcal D }(T_0^\gamma )\subset L^2(]-L,L[)\longrightarrow L^2(]-L,L[)\\ \varphi \longrightarrow T_0^\gamma \varphi = \sum \limits _{n=1}^{\infty }\lambda _n^{\gamma }\langle \varphi ,\varphi _n\rangle \varphi _n\\ \mathcal{D }(T_0^{\gamma })=\left\{ \varphi \in L^2(]-L,L[) \text{ such } \text{ that } \sum \limits _{n=1}^{\infty }\lambda _n^{2\gamma }\vert \langle \varphi ,\varphi _n\rangle \vert ^2<\infty \right\} \!\!. \end{array} \right. \end{aligned}$$

In the sequel, we consider the following operators:

$$\begin{aligned} \left\{ \begin{array}{l} B=T_0^{\frac{1}{2}}:\mathcal{D }(B)\subset L^2(]-L,L[)\longrightarrow L^2(]-L,L[)\\ {\varphi \longrightarrow B\varphi (x)=\frac{d^2\varphi }{dx^2}}\\ \mathcal{D }(B)=\left\{ \varphi \in L^2(]-L,L[) \text{ such } \text{ that } \sum \limits _{n=1}^{\infty }\lambda _n\vert \langle \varphi ,\varphi _n\rangle \vert ^2<\infty \right\} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} K:L^2(]-L,L[)\longrightarrow L^2(]-L,L[)\\ {\varphi \longrightarrow K\varphi (x)=\frac{i}{2}\int \limits _{-L}^L H_0(k\vert x-x^{\prime }\vert )\varphi (x^{\prime })dx^{\prime }}, \end{array} \right. \end{aligned}$$

and the following eigenvalue problem:

Find the values $\lambda (\varepsilon )\in \mathbb C$ for which there is a solution $u \in H_0^2(]-L,L[)\cap H^4(]-L,L[),\,u \ne 0,$ for the equation

$$\begin{aligned} T_0 u + \varepsilon K(T_0-B)u =\lambda (\varepsilon )(I+\varepsilon K)u, \end{aligned}$$

(6.6)

where $\lambda = \frac{m \omega ^2}{D}$ and $\varepsilon = \frac{2\rho _0}{m}.$

According to the definition given in reference [14, Chapter 9, Section 4], $\lambda$ is the eigenvalue and $u$ is the eigenmode. Note that $\lambda$ and $u,$ each one of them depends on the value of $\varepsilon .$ So, we denote $\lambda :=\lambda (\varepsilon )$ and $u:=u(\varepsilon ).$

For $\vert \varepsilon \vert <\frac{1}{\Vert K\Vert },$ the operator $I+\varepsilon K$ is invertible. So the problem (6.6) becomes:

Find the values $\lambda (\varepsilon )\in \mathbb C$ for which there is a solution $\varphi \in H_0^2(]-L,L[)\cap H^4(]-L,L[),\,\varphi \ne 0,$ for the equation

$$\begin{aligned} (I+\varepsilon K)^{-1}\frac{d^4\varphi }{dx^4}+\varepsilon (I+\varepsilon K)^{-1}K(T_0-B)\varphi =\lambda (\varepsilon )\varphi . \end{aligned}$$

The last problem is equivalent to: Find the values $\lambda (\varepsilon )\in \mathbb C$ for which there is a solution $\varphi \in H_0^2(]-L,L[)\cap H^4(]-L,L[),\,\varphi \ne 0,$ for the equation

$$\begin{aligned} \left( \frac{d^4}{dx^4}-\varepsilon K\frac{d^2}{dx^2}+ \varepsilon ^2 K^2\frac{d^2}{dx^2}+\cdots +(-1)^n\varepsilon ^n K^n\frac{d^2}{dx^2}+\cdots \right) \varphi =\lambda (\varepsilon )\varphi . \end{aligned}$$

We denote by $T_n:=(-1)^n K^n\frac{d^2}{dx^2},\,\mathcal{D }(T_n)=H^2(]-L,L[),$ for all $n\ge 1.$

Let us recall the following result:

Proposition 6.1

[4, Proposition   4.1] The following properties hold.

• (i) There exist constants $a,b,q>0$ and $\beta \in \left[ \frac{1}{2},\frac{3}{4}\right]$ such that

  $$\begin{aligned} \Vert T_k \varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert + b\Vert T_0 \varphi \Vert ^{\beta }\Vert \varphi \Vert ^{1-\beta } )\;\; {for\,all}\; \varphi \in \mathcal D (T_0)\,{and\,all}k\ge 1. \end{aligned}$$

• (ii) For $\vert \varepsilon \vert <\frac{1}{\Vert K \Vert },$ the series

  $$\begin{aligned} T(\varepsilon )\varphi :=T_0\varphi +\varepsilon T_1\varphi +\varepsilon ^2 T_2\varphi +\cdots +\varepsilon ^k T_k\varphi +\cdots \end{aligned}$$

converges for all $\varphi \in \mathcal D (T_0).$ $\square$

We denote by $T(\varepsilon ):=\frac{d^4}{dx^4}-\varepsilon K\frac{d^2}{dx^2}+ \varepsilon ^2 K^2\frac{d^2}{dx^2}+\cdots +(-1)^n\varepsilon ^n K^n\frac{d^2}{dx^2}+\cdots .$

The objective of this subsection is formulated as:

Theorem 6.1

There exist a sequence of complex $(\varepsilon _n)_n$ and a sequence $(\varphi _n(\varepsilon _n))_n$ of eigenvectors of $T(\varepsilon _n),$ which can be developed as entire series of $\varepsilon _n,$ such that the system $(\varphi _n(\varepsilon _n))_n$ forms a Schauder basis in $L^2(]-L,L[).$ $\square$

Proof

We have $T_0$ is self-adjoint and has compact resolvent. So, the system formed by its eigenvectors forms an orthonormal basis in $L^2(]-L,L[).$ Hence, the result follows immediately from Theorem 5.2 and Proposition 6.1. $\square$

6.2 Application to a Gribov operator in the Bargmann space

Reggeon field theory was introduced by Gribov [7] in $1967$ to study strong interactions, i.e., the interaction between protons and neutrons among other less stable particles. This theory is governed by a non self-adjoint Gribov operator [1, 7, 10] constructed as a polynomial in the standard annihilation operator $A$ and the standard creation operator $A^{*},$ defined in the Bargmann space

$$\begin{aligned} E=\left\{ \varphi :\mathbb C \longrightarrow \mathbb C \text{ entire } \text{ such } \text{ that } \displaystyle \int _\mathbb{C }e^{-\vert z\vert ^2}\vert \varphi (z)\vert ^2 dzd\bar{z}<\infty \text{ and } \varphi (0)=0\right\} \!\!. \end{aligned}$$

Let us consider the following Gribov operator:

$$\begin{aligned} (A^{*}A)^3+\varepsilon A^{*}(A+A^{*})A+\varepsilon ^2(A^{*}A)^{3u_2}+\cdots +\varepsilon ^k(A^{*}A)^{3u_k}+\cdots , \end{aligned}$$

(6.7)

where $\varepsilon \in \mathbb C$ and $(u_k)_{k\in \mathbb N }$ is a strictly decreasing sequence with strictly positive terms such that $u_0=1$ and $u_1=\frac{1}{2}.$

The Bargmann space $E$ is a Hilbert space for the scalar product $\langle .,.\rangle$ defined by

$$\begin{aligned} \left\{ \begin{array}{l} \langle .,.\rangle :{E}\times {E} \longrightarrow \mathbb C \\ \displaystyle {(\varphi , \psi ) \longrightarrow \langle \varphi , \psi \rangle }= \int _\mathbb{C }e^{-\vert z\vert ^2}\varphi (z) \bar{\psi }(z)dzd\bar{z}, \end{array} \right. \end{aligned}$$

and the associated norm is denoted by $\Vert ~.~\Vert .$

The annihilation operator $A$ and the creation operator $A^{*},$ are defined by:

$$\begin{aligned} \left\{ \begin{array}{l} A:\mathcal{D }(A)\subset {E}\longrightarrow {E}\\ \displaystyle {\varphi \longrightarrow A\varphi (z)=\frac{d\varphi }{d z}(z)}\\ \mathcal{D }(A)=\{\varphi \in {E} \text{ such } \text{ that } A\varphi \in {E}\} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} A^*:\mathcal{D }(A^*)\subset {E}\longrightarrow {E}\\ \displaystyle {\varphi \longrightarrow A^*\varphi (z)=z\varphi (z)}\\ { \mathcal D }(A^*)=\{\varphi \in {E} \text{ such } \text{ that } A^*\varphi \in {E}\}. \end{array} \right. \end{aligned}$$

Let $H_0$ and $H_1$ be the following operators:

$$\begin{aligned} \left\{ \begin{array}{l} H_0:{ \mathcal D }(H_0)\subset E\longrightarrow E\\ \displaystyle {\varphi \longrightarrow H_0\varphi (z)=A^{*}A\varphi (z)}\\ \mathcal{D }(H_0)=\{\varphi \in E \text{ such } \text{ that } H_0\varphi \in E\} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l} H_1:\mathcal{ D }(H_1)\subset E\longrightarrow E\\ \displaystyle {\varphi \longrightarrow H_1\varphi (z)=A^{*}(A+A^{*})A\varphi (z)}\\ \mathcal{ D }(H_1)=\{\varphi \in E \text{ such } \text{ that } H_1\varphi \in E\}. \end{array} \right. \end{aligned}$$

Remark 6.2

Due to [10, Lemme 3 p. 112], $H_0$ is a self-adjoint operator with compact resolvent. Moreover, $\{e_n(z)=\frac{z^n}{\sqrt{n!}}\}_{1}^{\infty }$ is a system of eigenvectors associated to the eigenvalues $\{n\}.$ So, the spectral decomposition of $H_0$ is given by:

$$\begin{aligned} H_0= \sum _{n=1}^{\infty }n\langle .,e_n\rangle e_n. \end{aligned}$$

Let $T_0=H_0^3.$ So, $T_0$ is defined by

$$\begin{aligned} \left\{ \begin{array}{l} T_0:{ \mathcal D }(T_0)\subset E\longrightarrow E\\ \displaystyle {\varphi \longrightarrow T_0\varphi =\displaystyle \sum _{n=1}^{\infty }n^3\langle \varphi ,e_n\rangle e_n}\\ { \mathcal D }(T_0)=\left\{ \varphi \in E \text{ such } \text{ that } \displaystyle \sum _{n=1}^{\infty }n^6\vert \langle \varphi ,e_n\rangle \vert ^2<\infty \right\} \!. \end{array} \right. \end{aligned}$$

$\square$

Using [10, Lemme 3, p. 112] and [10, Proposition 1, $(i)$ p. 103], we can see that:

Proposition 6.2

We have the following assertions:

• (i) $T_0$ is a closed linear operator with dense domain.

• (ii) The resolvent set of $T_0$ is not empty. In fact, $0\in \rho (T_0).$

• (iii) $T_0$ is a self-adjoint operator with compact resolvent.

• (iv) The eigenvalues of $T_0$ are simple and isolated.

• (v) The eigenvectors of $T_0$ form a Riesz basis in $E.$ $\square$

Let the operators $(T_0^{u_k})_{k\ge 0}:$

$$\begin{aligned} \left\{ \begin{array}{l} T_0^{u_k}:{ \mathcal D }(T_0^{u_k})\subset E\longrightarrow E\\ \displaystyle {\varphi \longrightarrow T_0^{u_k}\varphi =\displaystyle \sum _{n=1}^{\infty }n^{3u_k}\langle \varphi ,e_n\rangle e_n}\\ { \mathcal D }(T_0^{u_k})=\left\{ \varphi \in E \text{ such } \text{ that } \displaystyle \sum _{n=1}^{\infty }n^{6u_k}\vert \langle \varphi ,e_n\rangle \vert ^2<\infty \right\} \!. \end{array} \right. \end{aligned}$$

We can easily see that for all $k\ge 0\,\mathcal{D }(T_0^{u_k})\subset { \mathcal D }(T_0^{u_{k+1}}).$ So,

$$\begin{aligned} \bigcap _{k\ge 2}\mathcal{D }(T_0^{u_k})={ \mathcal D }(T_0^{u_2}). \end{aligned}$$

Let $\mathcal{D }=\mathcal{D }(T_0^{u_2})\bigcap \mathcal{D }(H_1)$ and $T_1$ (respectively, $(T_k)_{k\ge 2}$) the restriction of $H_1$ (respectively, $T_0^{u_k}$) to ${ \mathcal D }.$ Hence, the operators $(T_k)_{k\ge 1}$ have the same domain ${ \mathcal D }$ and we have ${ \mathcal D }(T_0)\subset { \mathcal D }.$

Let us denote by $T(\varepsilon ):=(A^{*}A)^3+\varepsilon A^{*}(A+A^{*})A+\varepsilon ^2(A^{*}A)^{3u_2}+\cdots +\varepsilon ^k(A^{*}A)^{3u_k}+\cdots$

The first result of this subsection is given by:

Proposition 6.3

There exist constants  $a, b, q>0$ and $\beta \in \left[ \frac{1}{2},1\right]$ such that

$$\begin{aligned} \Vert T_k\varphi \Vert \le q^{k-1}(a\Vert \varphi \Vert +b\Vert T_0\varphi \Vert ^\beta \Vert \varphi \Vert ^{1-\beta }) {for \,all}k\ge 1\,{and\,for\,all}\,\varphi \in \mathcal{D }(T_0). \end{aligned}$$

$\square$

Proof

First, we claim that there exists $c>0$ such that

$$\begin{aligned} \Vert T_k\varphi \Vert \le c\Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{1-u_k}\quad \text{ for } \text{ all } k\ge 1 \quad \text{ and } \text{ for } \text{ all }\quad \varphi \in \mathcal{D }(T_0). \end{aligned}$$

In fact,

$\bullet$ for $k=1\!\!:$

Let $\varphi \in \mathcal{D }(T_0).$ Due to [10], Remarque $4$ p. $111$], we have

$$\begin{aligned} T_1\varphi = \sum _{n=1}^{\infty }(n\sqrt{n+1}\langle \varphi , e_{n+1}\rangle +(n-1)\sqrt{n}\langle \varphi , e_{n-1}\rangle )e_n. \end{aligned}$$

So,

$$\begin{aligned} \Vert T_1\varphi \Vert =\left( \sum _{n=1}^{\infty }\left| n\sqrt{n+1}\langle \varphi , e_{n+1}\rangle +(n-1)\sqrt{n}\langle \varphi , e_{n-1}\rangle \right| ^2\right) ^{\frac{1}{2}}. \end{aligned}$$

Using Minkowski’s inequality, we obtain

$$\begin{aligned} \Vert T_1\varphi \Vert&\le \left( \sum _{n=1}^{\infty }\left| n\sqrt{n+1}\langle \varphi , e_{n+1}\rangle \right| ^2\right) ^{\frac{1}{2}}+\left( \sum _{n=1}^{\infty }\left| (n-1)\sqrt{n}\langle \varphi , e_{n-1}\rangle \right| ^2\right) ^{\frac{1}{2}}\\&\le \left( \displaystyle \sum _{n=1}^{\infty } n^2(n+1)\vert \langle \varphi , e_{n+1}\rangle \vert ^2\right) ^{\frac{1}{2}}+\left( \displaystyle \sum _{n=1}^{\infty }(n-1)^2n\vert \langle \varphi , e_{n-1}\rangle \vert ^2\right) ^{\frac{1}{2}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \Vert T_1\varphi \Vert \le c\left( \sum _{n=1}^{\infty }n^3\vert \langle \varphi , e_n\rangle \vert ^2\right) ^{\frac{1}{2}}, \quad \text{ for } \text{ all } \varphi \in \mathcal{D }(T_0), \end{aligned}$$

where

$$\begin{aligned} c=1+2\sqrt{2}. \end{aligned}$$

So,

$$\begin{aligned} \Vert T_1\varphi \Vert ^2\le c^2\Vert T_0^\frac{1}{2}\varphi \Vert ^2,\quad \text{ for } \text{ all } \varphi \in \mathcal{D }(T_0). \end{aligned}$$

(6.8)

On the other hand we have,

$$\begin{aligned} \Vert T_0^\frac{1}{2}\varphi \Vert ^2&= \langle T_0^\frac{1}{2}\varphi , T_0^\frac{1}{2}\varphi \rangle \\&= \langle T_0^\frac{1}{2}T_0^\frac{1}{2}\varphi , \varphi \rangle \\&= \langle T_0\varphi , \varphi \rangle \\&\le \Vert T_0\varphi \Vert \Vert \varphi \Vert . \end{aligned}$$

So, Eq. (6.8) implies that

$$\begin{aligned} \Vert T_1\varphi \Vert \le c\Vert T_0\varphi \Vert ^{\frac{1}{2}}\Vert \varphi \Vert ^{\frac{1}{2}}, \quad \text{ for } \text{ all } \varphi \in \mathcal{D }(T_0). \end{aligned}$$

(6.9)

$\bullet$ Now, let $k\ge 2$ and $\varphi \in \mathcal{D }(T_0).$ We have

$$\begin{aligned} T_k\varphi = \sum _{n=1}^{\infty }n^{3u_k}\langle \varphi ,e_n\rangle e_n. \end{aligned}$$

So,

$$\begin{aligned} \Vert T_k\varphi \Vert ^2&= \sum _{n=1}^{\infty }n^{6u_k}\vert \langle \varphi ,e_n\rangle \vert ^2\\&= \sum _{n=1}^{\infty }n^{6u_k}\vert \langle \varphi ,e_n\rangle \vert ^{2u_k}\vert \langle \varphi ,e_n\rangle \vert ^{2(1-u_k)}. \end{aligned}$$

Using Hölder’s inequality, we have

$$\begin{aligned} \Vert T_k\varphi \Vert ^2&\le \left( \sum _{n=1}^{\infty }n^6\vert \langle \varphi ,e_n\rangle \vert ^2\right) ^{u_k} \left( \sum _{n=1}^{\infty }\vert \langle \varphi ,e_n\rangle \vert ^2\right) ^{1-u_k}\\&\le \Vert T_0\varphi \Vert ^{2u_k}\Vert \varphi \Vert ^{2(1-u_k)}. \end{aligned}$$

So,

$$\begin{aligned} \Vert T_k\varphi \Vert \le \Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{1-u_k},\quad \text{ for } \text{ all } \varphi \in \mathcal{D }(T_0). \end{aligned}$$

(6.10)

Since $c>1,$ Eqs. (6.9) and (6.10) imply that for all $k\ge 1$

$$\begin{aligned} \Vert T_k\varphi \Vert \le c\Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{1-u_k},\quad \text{ for } \text{ all } \varphi \in \mathcal{D }(T_0). \end{aligned}$$

(6.11)

This ends the proof of the claim.

Second, let us recall an interpolation inequality: For $x, y, \delta$ and $\theta$ positive numbers such that $0\le \delta \le \theta ,$ then,

$$\begin{aligned} x^{\delta }y^{\theta -\delta }\le x^\theta +y^\theta . \end{aligned}$$

Let $\beta \in [u_1,u_0]$ and $\varphi \in \mathcal{D }(T_0).$ Writing $\Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{1-u_k}$ as

$$\begin{aligned} \Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{1-u_k}=\Vert \varphi \Vert ^{1-\beta }\Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{\beta -u_k}, \end{aligned}$$

and because for all $k\ge 1,~~0<u_k<\beta ,$ we can apply the foregoing interpolation inequality to $\Vert T_0\varphi \Vert ^{u_k}$ and $\Vert \varphi \Vert ^{\beta -u_k}$ with $\theta =\beta$ and $\delta =u_k,$ to deduce that

$$\begin{aligned} \Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{\beta -u_k}\le \Vert T_0\varphi \Vert ^{\beta }+\Vert \varphi \Vert ^{\beta }, \end{aligned}$$

and

$$\begin{aligned} \Vert T_0\varphi \Vert ^{u_k}\Vert \varphi \Vert ^{1-u_k}\le \Vert \varphi \Vert +\Vert T_0\varphi \Vert ^{\beta }\Vert \varphi \Vert ^{1-\beta }. \end{aligned}$$

Hence, Eq. (6.11) implies that for all $k\ge 1$

$$\begin{aligned} \Vert T_k\varphi \Vert \le c\Vert \varphi \Vert + c\Vert T_0\varphi \Vert ^{\beta }\Vert \varphi \Vert ^{1-\beta },\quad \text{ for } \text{ all } \varphi \in \mathcal{D }(T_0). \end{aligned}$$

So, it suffices to choose $a=b=c$ and $q=1$ and this achieves the proof of the result. $\square$

The objective of this subsection is formulated as:

Theorem 6.2

There exist a sequence of complex $(\varepsilon _n)_n$ and a sequence $(\varphi _n(\varepsilon _n))_n$ of eigenvectors of $T(\varepsilon _n),$ which can be developed as entire series of $\varepsilon _n,$ such that the system $(\varphi _n(\varepsilon _n))_n$ forms a Schauder basis in $E.$

Proof

We have already proved that the hypotheses (H1)–(H2) are fulfilled. Moreover, the eigenvectors $(e_n)_n$ of $T_0$ form an orthonormal basis in $E.$ So, the result follows from Theorem 5.2; which ends the proof of the theorem. $\square$