Abstract
In this note, by using gradients of Gateaux differentiable G-increasing functions we prove some refinements of the following Tam’s triangle-type inequality:
in the context of a compact connected Lie group G with Lie algebra \( {\mathfrak {g}} \) and corresponding Weyl chamber \( {\mathfrak {t}}_+ \). We also establish refinements of Tam’s inequality:
for a real semisimple Lie algebra \( {\mathfrak {g}} \) with Cartan decomposition \( {\mathfrak {g}} = {\mathfrak {k}} + {\mathfrak {p}} \), a maximal abelian subalgebra \( {{\mathfrak {a}}} \) in \( {\mathfrak {p}} \) and its closed Weyl chamber \( {{\mathfrak {a}}}_+ \).
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1 Motivation
We begin our presentation with notation and terminology quoted from [6].
Let G be a compact connected Lie group and \( {\mathfrak {g}} \) be its Lie algebra. Assume T is a maximal torus of G and \( {\mathfrak {t}} \) is the Lie algebra of T. By \( {\mathfrak {t}}_+ \), we denote a closed Weyl chamber in \( {\mathfrak {t}} \). For a given element \( x \in {\mathfrak {g}} \), the symbol \( {\mathfrak {t}}_+ (x) \) represents the unique element of the set \( {\mathfrak {t}}_+ \cap G x \), where \( G x = \{ g \cdot x : g \in G \} \) and \( g \cdot x = ( \mathrm{Ad}\,g ) x \).
Let \( \langle \cdot , \cdot \rangle \) be a G-invariant inner product on \( {\mathfrak {g}} \). The dual cone of \( {\mathfrak {t}}_+ \) is given by \( {\mathfrak {t}}_+^*= \{ x \in {\mathfrak {g}} : \langle x , v \rangle \ge 0 \, \text{ for } \text{ all }\, v \in {\mathfrak {t}}_+ \} \). This cone generates the preorder \( \le \) on \( {\mathfrak {g}} \) by \( y \le x \; \text{ iff } \; x - y \in {\mathfrak {t}}_+^*\) for \( x , y \in {\mathfrak {g}} \). In addition, a related preorder \( \prec \) can be defined by \( y \prec x \; \text{ iff } \; y \in \mathrm{conv}\,G x \) for \( x , y \in {\mathfrak {g}} \), where \( \mathrm{conv}\,G x \) is the convex hull of the G-orbit \( \{ g \cdot x : g \in G \} \) (cf. [3, Corollary B.3]). It is known that \( \le \) and \( \prec \) coincide on \( {\mathfrak {t}}_+ \) [1, Proposition 18].
In [6, Theorem 7] T.-Y. Tam presented a triangle-type inequality for connected compact groups, as follows
where \( {\mathfrak {t}}_+ ( z ) \) denotes the unique element in \( {\mathfrak {t}}_+ \cap G z \) corresponding to an element \( z \in {\mathfrak {g}} \).
A similar framework works in the context of real semisimple Lie algebras. Let \( {\mathfrak {g}} \) be a real semisimple Lie algebra with a Cartan decomposition \( {\mathfrak {g}} = {\mathfrak {k}} + {\mathfrak {p}} \), where \( {\mathfrak {p}} \ne 0 \). Let \( {{\mathfrak {a}}} \) be a maximal abelian subalgebra in \( {\mathfrak {p}} \) and \( {{\mathfrak {a}}}_+ \) be a closed Weyl chamber in \( {{\mathfrak {a}}} \).
The Killing form \( \langle \cdot , \cdot \rangle \) is positive definite on \( {\mathfrak {p}} \), so we can define the dual cone of \( {{\mathfrak {a}}}_+ \) by \( {{\mathfrak {a}}}_+^*= \{ x \in {\mathfrak {p}} : \langle x , v \rangle \ge 0 \; \text{ for } \text{ all } v \in {\mathfrak {p}} \}\). Then we introduce the preorder \( \le \) on \( {\mathfrak {p}} \) by \( y \le x \; \text{ iff } \; x - y \in {{\mathfrak {a}}}_+^*\) for \( x , y \in {\mathfrak {p}} \).
Let K be the maximal compact subgroup in the adjoint group \( \mathrm{Int}\,( {\mathfrak {g}} )\). So, \( \mathrm{Ad}\,K \) is maximal compact subgroup of \( \mathrm{Ad}\,G \). We define preorder \( \prec \) in the following manner: \( y \prec x \; \text{ iff } \; y \in \mathrm{conv}\,K x \) for \( x , y \in {\mathfrak {p}} \), where \( \mathrm{conv}\,K x \) is the convex hull of the K-orbit \( K x = \{ k \cdot x : k \in K \} \) with \( k \cdot x = ( \mathrm{Ad}\,k) x \) (cf. [3, Corollary B.3]). The preorders \( \le \) and \( \prec \) coincide on \( {{\mathfrak {a}}}_+ \) [2, Lemma 3.2]. For an element \( x \in {\mathfrak {p}} \), the symbol \( {{\mathfrak {a}}}_+ (x) \) represents the unique element of the set \( {{\mathfrak {a}}}_+ \cap K x \), where \( K x = \{ k \cdot x : k \in K \} \) and \( k \cdot x = ( \mathrm{Ad}\,k ) x \).
In [6, Theorem 2] T.-Y. Tam showed that
In this note, our purpose is to show refinements of inequalities (1) and (2) by employing gradients of differentiable real functions increasing with respect to corresponding preorder \( \prec \) (cf. [6]).
2 Results for Compact Connected Lie Groups
In this section, we consider the Lie algebra \( {\mathfrak {g}} \) of a compact connected Lie group G. We use the norm \( \Vert {\cdot }\Vert = \langle \cdot , \cdot \rangle ^{1/2} \) on \( {\mathfrak {g}} \) generated by a G-invariant inner product \( \langle \cdot , \cdot \rangle \) on \( {\mathfrak {g}} \). The group G acts on \( {\mathfrak {g}} \) by \( \mathrm{Ad}\,G \). The preorder \( \prec \) is defined in Sect. 1.
A function \( \varPhi \) defined on \( {\mathfrak {g}} \) is called G-invariant if
We say that a function \( \varPhi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) is G-increasing, if for \( x , y \in {\mathfrak {g}} \),
In order to state our results, we need to employ the convex cone \( {\mathcal C} \) of all G-increasing real functions defined on \( {\mathfrak {g}} \) as well as the cone preorder \( \le _{\mathcal C} \) generated by \( {\mathcal C} \), as follows: given two real functions \( \varphi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) and \( \psi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\), we use the notation \( \psi \le _{\mathcal C} \varphi \) whenever the difference function \( \varphi - \psi \) is G-increasing on \( {\mathfrak {g}} \).
Given two vectors \( x , y \in {\mathfrak {g}} \), if there exists a \( g \in G \) such that \( x \in g \, {\mathfrak {t}}_+ \) and \( y \in g \, {\mathfrak {t}}_+ \), then
In this note, we take the convention that the Gateaux differentiability of a function \( \varPhi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) means the existence of the directional derivative
at each point \( y \in {\mathfrak {g}} \) and in each direction \( h \in {\mathfrak {g}} \), and moreover that the map \( {\mathfrak {g}} \ni h \rightarrow \nabla _h \varPhi (y) \in {\mathbb {R}}\) is continuous and linear as a function of h. Consequently, there exists the gradient \( \nabla \varPhi (y) \in {\mathfrak {g}} \) satisfying the condition
In accordance with [4, Theorem 2.1], a Gateaux differentiable G-increasing function \( \varPhi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) with continuous gradient \( \nabla \varPhi (\cdot ) \) satisfies the condition
Theorem 1
Let \( \varphi \) and \( \psi \) be Gateaux differentiable real functions on \( {\mathfrak {g}} \) with continuous gradients \( \nabla \varphi (\cdot )\) and \( \nabla \psi (\cdot )\), respectively. Suppose that
i.e., the functions \( \psi \), \( \varphi - \psi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are G-increasing on \( {\mathfrak {g}} \).
If \( x , y \in {\mathfrak {g}} \) then
Remark 1
Observe that for any points \( x , y \in {\mathfrak {g}} \), statement (8) in Theorem 1 shows the anti-isotonity of the functional
with \( \phi \) running over the set of all Gateaux differentiable real functions defined on \( {\mathfrak {g}} \). Here the anti-isotonity is with respect to the pair \( ( \le _{\mathcal C} , \prec ) \) on the set
Proof
By making use of (1), we obtain
The preorder \( \prec \) restricted to \( {\mathfrak {t}}_+ \) is a cone preorder on \( {\mathfrak {t}}_+ \), so we have
For the element \( y \in {\mathfrak {g}} \), there exists a \( g \in G \) such that \( y = g \cdot {\mathfrak {t}}_+ ( y ) \in g \, {\mathfrak {t}}_+ \). Because of the G-increase in the functions \( \psi \) and \( \varphi - \psi \), it follows from (6) that
[see (6)]. This via (3) ensures that
In consequence, we get
Since
and the functions \( \varphi = \psi + ( \varphi - \psi ) \) and \( \nabla \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are G-increasing, we deduce from (6) that
and therefore by (3) we have
and next
Similarly, the function
is G-increasing, because \( \psi \), \( \varphi - \psi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are so. Hence, via (6) we have
This and (3) yield
Finally, by (11)–(13) and (10) we obtain
as claimed. \(\square \)
As a corollary to Theorem 1, we now present a refinement of the triangle-type inequality (1).
Theorem 2
Let \( \varphi \) be a Gateaux differentiable real function on \( {\mathfrak {g}} \) with continuous gradient \( \nabla \varphi (\cdot )\). Suppose that
i.e., the functions \( \varphi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are G-increasing on \( {\mathfrak {g}} \).
If \( x , y \in {\mathfrak {g}} \) then
Proof
It follows from (14) that
with \( \psi = 0 \). By applying Theorem 1 with \( \nabla \psi (y) = 0 \), we obtain
By virtue of (14), we get
with \( \psi = \frac{1}{2} \Vert {\cdot }\Vert ^2 \). By using Theorem 1 with \( \nabla \psi (y) = y \), we establish
This completes the proof.\(\square \)
Corollary 1
Let \( 0 \le t \le 1 \). If \( x , y \in {\mathfrak {g}} \) then
Proof
The function \( \Vert {\cdot }\Vert = \langle \cdot , \cdot \rangle ^{1/2} \) is G-increasing on \( {\mathfrak {g}} \), because it is convex and G-invariant. Therefore, the functions \( \frac{1}{2} \Vert {\cdot }\Vert ^2 \), \( \varphi = t \frac{1}{2} \Vert {\cdot }\Vert ^2 \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi = ( 1 - t ) \frac{1}{2} \Vert {\cdot }\Vert ^2 \) are G-increasing on \( {\mathfrak {g}} \). That is,
So, by Theorem 2 with \( \nabla \varphi (y) = t y \), we infer that inequality (16) holds valid.\(\square \)
3 Results for Real Semisimple Lie Algebras
In this section, we show corresponding results to those in Sect. 2 for a real semisimple Lie algebra \( {\mathfrak {g}} \) with a Cartan decomposition \( {\mathfrak {g}} = {\mathfrak {k}} + {\mathfrak {p}} \), \( {\mathfrak {p}} \ne 0 \), and with a maximal abelian subalgebra \( {{\mathfrak {a}}} \) in \( {\mathfrak {p}} \) equipped with a fixed closed Weyl chamber \( {{\mathfrak {a}}}_+ \) in \( {{\mathfrak {a}}} \). The inner product \( \langle \cdot , \cdot \rangle \) on \( {\mathfrak {p}} \) is the restriction of Killing form, and as previously the norm is given by \( \Vert {\cdot }\Vert = \langle \cdot , \cdot \rangle ^{1/2} \). The preorder \( \prec \) is defined in Sect. 1.
The maximal compact subgroup in the adjoint group \( \mathrm{Int}\,( {\mathfrak {g}} )\) is denoted by K. It acts on \( {\mathfrak {p}} \) by \( \mathrm{Ad}\,K \), i.e., \( k \cdot x = ( \mathrm{Ad}\,k ) x \) for \( k \in K \) and \( x \in {\mathfrak {p}} \).
A function \( \varPhi \) defined on \( {\mathfrak {p}} \) is called K-invariant if
We say that a function \( \varPhi : {\mathfrak {p}} \rightarrow {\mathbb {R}}\) is K-increasing, if for \( x , y \in {\mathfrak {p}} \),
Here, by definition, the convex cone \( {\mathcal C} \) consists of all K-increasing real functions defined on \( {\mathfrak {p}} \). The cone preorder \( \le _{\mathcal C} \) induced by \( {\mathcal C} \) is defined as follows: for any two real functions \( \varphi : {\mathfrak {p}} \rightarrow {\mathbb {R}}\) and \( \psi : {\mathfrak {p}} \rightarrow {\mathbb {R}}\), we write \( \psi \le _{\mathcal C} \varphi \) provided that the difference function \( \varphi - \psi \) is K-increasing on \( {\mathfrak {p}} \).
Theorem 3
Let \( \varphi \) and \( \psi \) be Gateaux differentiable real functions on \({\mathfrak {p}}\) with continuous gradients \( \nabla \varphi (\cdot )\) and \( \nabla \psi (\cdot )\), respectively. Suppose that
i.e., the functions \( \psi \), \( \varphi - \psi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are K-increasing on \( {\mathfrak {p}} \).
If \( x , y \in {\mathfrak {p}} \) then
Proof
The proof of Theorem 3 is similar to that of Theorem 1, and therefore omitted.\(\square \)
An analog of Theorem 2 is the following.
Theorem 4
Let \( \varphi \) be a Gateaux differentiable real function on \( {\mathfrak {p}} \) with continuous gradient \( \nabla \varphi (\cdot )\). Suppose that
i.e., the functions \( \varphi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are K-increasing on \( {\mathfrak {p}} \).
If \( x , y \in {\mathfrak {p}} \) then
Proof
Use a similar method as in the proof of Theorem 2.\(\square \)
Finally, we present an analog of Corollary 1.
Corollary 2
Let \( 0 \le t \le 1 \). If \( x , y \in {\mathfrak {p}} \) then
Proof
It follows easily from Theorem 4 applied to the function \( \varphi = t \frac{1}{2} \Vert {\cdot }\Vert ^2 \) with \( \nabla \varphi (y) = t y \).\(\square \)
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Communicated by Rosihan M. Ali.
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Niezgoda, M. Refinements of Triangle-Like Inequalities in Lie’s Framework. Bull. Malays. Math. Sci. Soc. 44, 243–250 (2021). https://doi.org/10.1007/s40840-020-00955-2
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DOI: https://doi.org/10.1007/s40840-020-00955-2