Abstract
Rochberg’s coboundary theorem provides conditions under which the equation \((I-T)y = x\) is solvable in y. Here T is a unilateral shift on Hilbert space, I is the identity operator and x is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by T and growth of iterates of T at x. We revisit Rochberg’s theorem and prove the following result. Let T be an isometry acting on a Hilbert space \(\mathcal H\) and let \(x \in \mathcal H\). Suppose that \( \sum _{k=0}^\infty k \Vert T^{*k} x \Vert < \infty . \) Then x is in the range of \((I-T)\) if (and only if) \(\left\Vert\sum _{k= 0}^n T^k x \right\Vert = o(\sqrt{n}).\) When T is merely a contraction, x is a coboundary under an additional assumption. Some applications to \(L^2\)-solutions of the functional equation \(f(x)-f(2x) = F(x)\), considered by Fortet and Kac, are given.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 Coboundaries
Let T be a bounded linear operator acting on a complex Banach space \(\mathcal X\). An element x of \(\mathcal X\) is called a coboundary for T if there is \(y\in \mathcal X\) such that \(x = y - Ty\). Coboundaries are related to the behavior of the ergodic sums
A variant of the mean ergodic theorem for power bounded operators on reflexive Banach spaces has been proved by von Neumann for Hilbert spaces and by Lorch in the general case; see for instance [14]. Recall that T is said to be power bounded if \(\sup _{n\ge 1} \Vert T^n\Vert < \infty \). We have
In particular, as a consequence of this ergodic decomposition, we have
One can say more about the rate of convergence of \((1/n)S_n(T) x\) to zero when x is a coboundary. Indeed, when there exists a solution y of the equation \(y - Ty = x\), the ergodic sums satisfy \(S_n(T) x = y - T^ny\). It follows that \((S_n(T)x)_{n\in \mathbb {N}}\) is bounded. Therefore
This rate of convergence to zero, namely O(1/n), characterizes coboundaries of power bounded operators on reflexive spaces. Indeed, the converse result (whenever T is power bounded and \(\mathcal X\) is reflexive, an element x satisfying (1.1) is a coboundary for T) has been proved by Browder [1] and rediscovered by Butzer and Westphal [2].
We also note (see for instance [3, 4, 11] and the references therein) that if \((I-T) \mathcal X\) is not closed, then for every sequence \((a_n)_{n\ge 1}\) of positive real numbers converging to zero, there exists \(x \in \overline{(I-T)\mathcal X} \backslash (I-T)\mathcal X\) such that
In particular, there is no general rate of convergence in the mean ergodic theorem outside coboundaries.
1.2 Rochberg’s Theorem
Browder’s theorem has been extended to the case that T is a dual operator on a dual Banach space by Lin [16]; see also Lin and Sine [17]. We refer the reader to the introduction of [6], and the references cited therein, for the history of Browder’s theorem and for other extensions and generalizations. We mention here only two references, namely [19] and [13], dealing with the Hilbert space situation. Any of these Hilbert or Banach space abstract characterizations is not strong enough to obtain as consequences classical results of Fortet and Kac [9, 12] who dealt with the case \(\mathcal X= L^2(0,1)\) and \(Sf(x) = f(2x)\). This operator S is the Koopman operator associated with the doubling map on the torus; see the last section of this manuscript for more information about coboundaries of S. This situation has been remedied by Rochberg [20], who showed that a condition of \(o(\sqrt{n})\) growth of ergodic sums at x is sufficient to ensure that x is a coboundary for a unilateral shift on Hilbert space. Notice that the Koopman operator S acts as a unilateral shift on the subspace of \(L^2 (0,1)\) of functions whose zeroth Fourier coefficient vanishes.
We need the following classical definition in order to state Rochberg’s abstract coboundary theorem.
Definition 1.1
Let T be an isometry acting on Hilbert space \(\mathcal H\). A closed subspace \(\mathcal K\) of \(\mathcal H\) is called wandering for T whenever
The isometry T is called a (unilateral) shift if \(\mathcal H\) possess a closed subspace \(\mathcal K\), wandering for T and such that
Theorem 1.2
([20]) Let S be a shift and let f be an element of \(\mathcal H\). Using the notation of the preceding definition, we denote by \(f_j\) the projection of f onto the closed subspace \(S^j \mathcal K\). Suppose that there exists \(\beta > 0\) such that
Then there exists g in \(\mathcal H\) such that \((I-S)g = f\) if and only if
Remark 1.3
The condition
is of course dependent of the decomposition of \(\mathcal H\) associated with the unilateral shift S. It implies \(\Vert S^{* j} f \Vert = O (2^{-\beta j})\).
1.3 Statement of the Main Results
In the next theorem the unilateral shift S is replaced by an arbitrary isometry T and the growth of the norm of the projection \(f_j\) by the convergence of the series \(\sum _{j=0}^\infty j\Vert T^{* j} f \Vert \). The statement of the result does not depend on the Wold decomposition, at least not in an explicit way. For the convenience of the reader, the Wold decomposition theorem is recalled below. Theorem 1.4 implies Rochberg’s theorem and it allows to recover Kac’s results about the coboundaries of the Koopman operator of the doubling map.
Theorem 1.4
Let T be an isometry acting on a Hilbert space \(\mathcal H\) and let \(x \in \mathcal H\). Suppose that
Then there exists \(y \in \mathcal H\) such that \(x = (I-T)y\) if and only if
Note however that the condition (1.2) implies that x is necessarily an element of the shift part of the isometry T.
Considering coboundaries of adjoints of isometries, we notice that the identity \(I-T = (T^*-I)T\) shows that every coboundary of the isometry T is also a coboundary for its adjoint \(T^*\). It follows from [7, Proposition 4.3] that when the isometry T is not invertible (i.e., not a unitary operator), there are coboundaries for \(T^*\) which are not coboundaries for T.
The following result, more general than Theorem 1.4, is about coboundaries of contractions (operators of norm no greater than one).
Theorem 1.5
Let T be a linear operator acting on a Hilbert space \(\mathcal H\) with \(\Vert T\Vert \le 1\). Let \(x \in \mathcal H\) and denote \(S_n(T) x := x + Tx + \dots + T^{n-1}x\). Suppose that (1.2) holds, as well as
and
Then there exists \(y \in \mathcal H\) such that \(x = (I-T)y\). In addition, y can be chosen such that \(\Vert Ty\Vert = \Vert y\Vert \).
We obtain the following consequence.
Corollary 1.6
Let T be a linear operator acting on a Hilbert space \(\mathcal H\) with \(\Vert T\Vert \le 1\). Let \(x \in \mathcal H\) and denote \(S_n(T) x := x + Tx + \dots + T^{n-1}x\). Suppose that (1.2) and (1.3) hold, as well as
Then there exists \(y \in \mathcal H\) such that \(x = (I-T)y\) and \(\Vert Ty\Vert = \Vert y\Vert \).
Some remarks are in order. Theorem 1.5 and its consequence Corollary 1.6 show that the coboundary equation can be solved within the maximal isometric subspace
We refer to [18] and [15] for the canonical decomposition of a contraction into the maximal isometric subspace and its orthogonal.
Conditions (1.4) and (1.5) are easily verified when T is an isometry. The conditions (1.2) and (1.3) are always satisfied when \(\Vert T\Vert < 1\); however (1.5) is not, unless \(x=0\). In fact, \(\Vert Ty\Vert = \Vert y\Vert \) and \(\Vert T\Vert < 1\) imply that \(y=0\) and thus \(x=0\). Of course, as \((I-T)\) is invertible when \(\Vert T\Vert < 1\) by Carl Neumann’s lemma, the coboundary equation \(x = (I-T)y\) is always solvable in this case.
1.4 Outline of the Paper
A proof of Theorem 1.4 is given in the next section. The more general Theorem 1.5 and its consequence Corollary 1.6 are proved in Sect. 3. Some applications to the functional equation \(g(x) - g(2x) = f(x)\) are presented in the next section. The last section collects the acknowledgments, and (imposed) conflict of interest and data availability statements.
2 Proof of Theorem 1.4
We first recall Wold’s decomposition Theorem (see [18, Chapter 1]).
Theorem 2.1
(Wold decomposition) Let T be an isometry on a Hilbert \(\mathcal H\). Then \(\mathcal H\) decomposes as an orthogonal sum \(\mathcal H= \mathcal H_0 \oplus \mathcal H_1\) such that \(\mathcal H_0\) and \(\mathcal H_1\) are reducing for T, the restriction of T to \(\mathcal H_0\) is a unitary operator and the restriction of T to \(\mathcal H_1\) is a unilateral shift (one of the subspaces can eventually reduce to \(\{0\}\)). This decomposition is unique; in particular, we have
Proof of Theorem 1.4
If \(x = (I-T) y\), then \(\sum _{k= 0}^n T^k x = x - T^{n+1}x\). Therefore \(\sum _{k= 0}^n T^k x\) is bounded since the isometry T is clearly power-bounded. In particular,
Suppose now that
We want to show the existence of a solution y of the equation \((I-T)y = x\).
Let \(\mathcal H= \mathcal H_0 \oplus \mathcal H_1\) be the Wold’s decomposition associated with T. We notice that \(x \in \mathcal H_1\). Indeed, if \(x = x_0 + x_1\) according to Wold’s decomposition of \(\mathcal H\), then
Therefore
On the other hand, it follows from (1.2) that
We obtain that \(x \in \mathcal H_1\). In particular, if \(\mathcal H_1\) is reduced to \(\{0\}\), then \(x = 0 = (I-T) 0\). Therefore, without loss of any generality, we can assume that T is a shift.
For each \(n \in \mathbb {N}\), we denote by \(P_n\) the projection onto the subspace \(T^n \mathcal K\). For \(u \in \mathcal H\), we set \(u_n:=P_n(u)\), \(u^n: = \sum _{j=0}^n u_j\) and \(R_n := u-u^n\).
Suppose that y is solution of the equation \((I-T)y = x\). We first obtain, by projecting to \(T^k \mathcal K\) for each \(k \in \mathbb {N}\), the following system of equations :
We then obtain
Consider now, for each \(r \in \mathbb {N}\), the element
We will prove that \(\sum _{r=0}^\infty \Vert y_r\Vert ^2\) is convergent, thus showing that \(y = \sum _{r=0}^\infty y_r\) is well defined in \(\mathcal H\). In that case, for every \(r\in \mathbb {N}\), we have
This shows that \((I-T)y = x\).
To prove that \(\sum _{r=0}^\infty \Vert y_r\Vert ^2\) is finite, we need two more results.
Lemma 2.2
Let \(u\in \mathcal H\) be such that \(\sum _{j\ge 0} \Vert T^{*j}u \Vert < +\infty \). Then
Proof
We first notice that the sum \(\sum _{k=1}^\infty \langle u; T^k u \rangle \) is absolutely convergent since \(( \Vert T^{*j}u \Vert )_{j\ge 0}\) is summable. For each \(n \in \mathbb {N}^*\), we have
On the other hand, we have
Using again the summability of the sequence \((\Vert T^{*j}u \Vert )_{j\ge 0}\) and the Kronecker’s lemma (see for instance [21, Lemma IV.3.2]), we get
As the series \(\sum _{k\ge 1} \langle u; T^k u \rangle \) is convergent, we obtain, as n tends to infinity,
\(\square \)
Lemma 2.3
Let \(u \in \mathcal H\). For every \(r \in \mathbb {N}\) we have
Proof
Let \(n \ge r\). For \(k \in \mathbb {N}\) we have
Using the decomposition of \(\mathcal H\) as \(\mathcal H= \bigoplus _{n=0}^\infty T^n \mathcal K\), we obtain
We have
and
as well as
We thus obtain
\(\square \)
We finally show that \(\sum _{r \ge 0} \Vert y_r\Vert ^2 < \infty \). Using Lemma 2.3, we have for each \(r \in \mathbb {N}\),
Using the parallelogram identity for the vectors \(x^r + R_r = x\), we get
Make now n tends to infinity. Using Lemma 2.2 for \(R_r\) and \(x^r - R_r\), and the hypothesis \(\frac{1}{n} \left\| \sum _{k=0}^n T^k x \right\| ^2 \underset{ n\rightarrow \infty }{\longrightarrow } 0\), we obtain
Using now Lemma 2.2 applied to \(x^r\) and Lemma 2.3, we get
We can infer that
so
For each fixed r we have \(R_r = T^{r+1} T^{*(r+1)}x\). Thus \( \Vert R_r\Vert = \Vert T^{*(r+1)}x\Vert \). As
we obtain that \((\Vert R_r\Vert ^2)_r\) is summable. It suffices to show that
We have
Using again the summability of \((r \Vert T^{*r}x\Vert )_r\), we get
Therefore \(\sum _{r=0}^\infty \Vert y_r\Vert ^2 <\infty .\) \(\square \)
3 The Case of Contractions
We now prove Theorem 1.5 and its consequence Corollary 1.6.
Proof of Theorem 1.5
Let D denote the defect operator \(D = (I-T^*T)^{1/2}\), which is well defined since T is a contraction. As
the operator \(R : \ell ^2(\mathcal H) \mapsto \ell ^2(\mathcal H)\) given by
and with matrix representation
is an isometry. We can thus apply Theorem 1.4 to R.
The iterates of R are given by
while their adjoints are given by
Denote \(\tilde{x} = (x,0,0, \cdots ) \in \ell ^2(\mathcal H)\) and \(\tilde{y} = (y,y_1,y_2, \cdots ) \in \ell ^2(\mathcal H)\). The equation
reduces to the system of equations \(x = (I-T)y\), \(y_1 = Dy\), \(y_2 = y_1\), \(y_3 = y_2\), etc. As \(\tilde{y} \in \ell ^2(\mathcal H)\), we obtain \(y_1 = y_2 = \cdots = 0\). Therefore the equation \(\tilde{x} = (I-R)\tilde{y}\) in \(\ell ^2(\mathcal H)\) is equivalent to
Every positive (i.e. positive semi-definite) operator has the same kernel as its positive square-root; thus \((I-T^*T)y = 0\). Therefore \(\Vert Ty\Vert = \Vert y\Vert \).
An easy computation shows that the summability condition \(\sum _{k=0}^\infty k \Vert R^{*k} \tilde{x} \Vert < \infty \) is equivalent to \(\sum _{k=0}^\infty k \Vert T^{*k}x \Vert < \infty \).
Notice now that
Therefore
Hence, using the notation \(S_n(T) x = x + Tx + \dots + T^{n-1}x\), the \(o(\sqrt{n})\) condition
is equivalent to
The proof is now complete using the identity \(\Vert Du\Vert ^2 = \Vert u\Vert ^2 - \Vert Tu\Vert ^2\). \(\square \)
Corollary 1.6 follows from Theorem 1.5 and Kronecker’s lemma, already used in the proof of Theorem 1.4.
4 Coboundaries of the Doubling Map
Let \({{\,\textrm{val}\,}}_2(n)\) be the 2-valuation of n, that is
For \(n\in \mathbb {Z}\), we denote by \( \hat{f}(n) = \int _0^1 f(t)e^{-int} \, dt\) the n-th Fourier coefficient of \(f \in L^2 (0,1)\).
Corollary 4.1
Suppose f is a periodic function of period 1 such that \(f \in L^2 (0,1)\),
and there exists \(\varepsilon >0\) such that
Then there is a function g in \(L^2(0,1)\) of period one such that
if and only if
Proof
We use Theorem 1.4 applied to the isometry \(T : L^2(0,1) \longrightarrow L^2(0,1)\) defined by
We first remark that condition (4.1) is justified by the fact that T acts as a shift operator on the subspace of \(L^2 (0,1)\) of functions whose zeroth Fourier coefficient vanishes. The condition
is exactly the condition
which appears in Theorem 1.4. We want to show that
Recall that T acts as a shift operator on the subspace of \(L^2 (0,1)\) of functions whose zeroth Fourier coefficient vanishes. Let \((a_n) = (\hat{f}(n))\) be the sequence of Fourier coefficients of f. We have \(a_0 = 0\). The iterates of the adjoint of T at f can be computed as
For \(\varepsilon >0\), using the change \(n = j2^k\) in the order of summation, we get
Thus, under our hypothesis about the Fourier coefficients, we have
\(\square \)
Corollary 4.2
[20] Let f be a periodic function of period 1 such that \(f \in L^2 (0,1)\),
and there exists \(\alpha > 0\) such that
Then there is a function g in \(L^2(0,1)\) of period one such that
if and only if
Proof
The result follows from Corollary 4.1 with \(\varepsilon = 1\), say. Indeed, using the condition (4.3), one can estimate
\(\square \)
Remark 4.3
Condition (4.3) is condition (a) from Theorem 4 in [20]. It has been proved in [20] that each of other three conditions of Hölder type, called there (b), (c) and (d), implies the condition (4.3). Mark Kac has already considered in [12] the case when f is in the Hölder class \(C^{0,\alpha }\) for some \(\alpha > 1/2\). We refer to [5, 9, 10] for other contributions concerning the functional equation \(f(t) = g(t) - g(2t)\).
Remark 4.4
All the remarks at the end of the paper [20] apply also in our situation. In particular, the generalization to the functional equation \(f(t) = g(t) - g(nt)\) (for a fixed integer n) is immediate.
Data Availability
No datasets were generated or analysed during the current study.
References
Browder, F.: On the iteration of transformations in noncompact minimal dynamical systems. Proc. Amer. Math. Soc. 9, 773–780 (1958)
Butzer, P.L., Westphal, U.: The mean ergodic theorem and saturation. Indiana Univ. Math. J. 20, 1163–1174 (1970/1971)
Badea, C., Müller, V.: On weak orbits of operators. Topology Appl. 156, 1381–1385 (2009)
Badea, C., Grivaux, S., Müller, V.: The rate of convergence in the method of alternating projections. Algebra i Analiz 23 (2011), no. 3, 1–30; reprinted in St. Petersburg Math. J. 23 (2012), no. 3, 413–434
Ciesielski, Z.: On the functional equation \(f(t) = g(t) - g(2t)\). Proc. Amer. Math. Soc. 13, 388–392 (1962)
Cohen, G., Lin, M.: Double coboundaries for commuting contractions. Pure Appl. Funct. Anal. 2, 11–36 (2017)
Derriennic, Y., Lin, M.: Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123, 93–130 (2001)
Devys, O.: Localisation spectrale à l’aide de polynômes de Faber et équation de cobord, Ph.D. thesis, Université de Lille, (2012)
Fortet, R.: Sur une suite également répartie. Studia Math. 9, 54–69 (1940)
Fukuyama, K.: On a gap series of Mark Kac. Colloq. Math. 81(2), 157–160 (1999)
Gomilko, A., Haase, M., Tomilov, Y.: On rates in mean ergodic theorems. Math. Res. Lett. 18(2), 201–213 (2011)
Kac, M.: On the distribution of values of sums of the type \(\sum f(2^k t)\). Ann. of Math. (2) 47, 33–49 (1946)
Kozma, G., Lev, N.: Exponential Riesz bases, discrepancy of irrational rotations and BMO. J. Fourier Anal. Appl. 17(5), 879–898 (2011)
Krengel, U.: Ergodic theorems. With a supplement by Antoine Brunel. de Gruyter Studies in Mathematics, 6. Walter de Gruyter and Co., Berlin. viii+357 pp. ISBN: 3-11-008478-3 (1985)
Levan, N.: Canonical decompositions of completely nonunitary contractions. J. Math. Anal. Appl. 101, 514–526 (1984)
Lin, M.: On quasi-compact Markov operators. Ann. Probab. 2, 464–475 (1974)
Lin, M., Sine, R.: Ergodic theory and the functional equation \((I-T)x = y\). J. Oper. Theory 10, 153–166 (1983)
Sz.-Nagy, B., Foiaş, C., Bercovici, H., Kérchy, L.: Harmonic analysis of operators on Hilbert space. Second edition. Revised and enlarged edition. Universitext. Springer, New York, 2010 (First edition published in 1967)
Robinson, E.: Sums of stationary random variables. Proc. Amer. Math. Soc. 11, 77–79 (1960)
Rochberg, R.: The equation \((I-S)g=f\) for shift operators in Hilbert space. Proc. Amer. Math. Soc. 19, 123–129 (1968)
Shiryaev, A.N.: Probability. 2. Third edition. Translated from the 2007 fourth Russian edition by R. P. Boas and D. M. Chibisov. Graduate Texts in Mathematics, 95. Springer, New York, (2019)
Zygmund, A.: Trigonometric series. Vol. I, II. Third edition. With a foreword by Robert A. Fefferman. Cambridge Mathematical Library. Cambridge University Press, Cambridge, (2002). xii; Vol. I: xiv+383 pp.; Vol. II: viii+364 pp. ISBN: 0-521-89053-5
Acknowledgements
Some of the results presented here are part of the 2012 PhD thesis of the second-named author [8] written under the supervision of the first-named author. We wish to thank several persons who encouraged us to present these results to a larger audience and/or to revisit them. As Amor Towles said: “For as it turns out, one can revisit the past quite pleasantly, as long as one does so expecting nearly every aspect of it to have changed”. Special thanks are due to Michael Lin for several interesting comments and remarks. We would like to thank the anonymous referee for a careful reading of the manuscript and very useful suggestions. The first-named author would like to thank the Max Planck Institute for Mathematics in Bonn for providing excellent working conditions and support.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Communicated by Mihai Putinar.
To the memory of Jörg Eschmeier.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021), by the Labex CEMPI (ANR-11-LABX-0007-01) and by the Max Planck Institute of Mathematics (Bonn).
This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz, Mihai Putinar and Ernst Albrecht.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Badea, C., Devys, O. Rochberg’s Abstract Coboundary Theorem Revisited. Complex Anal. Oper. Theory 16, 115 (2022). https://doi.org/10.1007/s11785-022-01293-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-022-01293-w