Some Operator Bounds Employing Complex Interpolation Revisited

TL;DR

This paper revisits and extends bounds on operator-valued functions using complex interpolation, focusing on self-adjoint and sectorial operators, and derives inequalities involving bounded and trace class operators.

Contribution

It provides new bounds on operator functions employing complex interpolation, generalized polar decomposition, and Heinz's inequality, especially for sectorial operators with bounded imaginary powers.

Findings

Established bounds for operator-valued functions using complex interpolation.

Derived inequalities for operators in trace ideals.

Extended known bounds to more general classes of operators.

Abstract

We revisit and extend known bounds on operator-valued functions of the type $$ T_1^{-z} S T_2^{-1+z}, \quad z \in \ol \Sigma = \{z\in\bbC\,|\, \Re(z) \in [0,1]\}, $$ under various hypotheses on the linear operators $S$ and $T_j$, $j=1,2$. We particularly single out the case of self-adjoint and sectorial operators $T_j$ in some separable complex Hilbert space $\cH_j$, $j=1,2$, and suppose that $S$ (resp., $S^*$) is a densely defined closed operator mapping $\dom(S) \subseteq \cH_1$ into $\cH_2$ (resp., $\dom(S^*) \subseteq \cH_2$ into $\cH_1$), relatively bounded with respect to $T_1$ (resp., $T_2^*$). Using complex interpolation methods, a generalized polar decomposition for $S$, and Heinz's inequality, the bounds we establish lead to inequalities of the following type, \begin{align*} & \big\|\ol{T_2^{-x}ST_1^{-1+x}}\big\|_{\cB(\cH_1,\cH_2)} \leq N_1 N_2 e^{(\theta_1 + \theta_2) [x(1-x)]^{1/2}} \\ & \quad \times \big\|ST_1^{-1}\big\|_{\cB(\cH_1,\cH_2)}^{1-x} \, \big\|S^*(T_2^*)^{-1}\big\|_{\cB(\cH_2,\cH_1)}^{x}, \quad x \in [0,1], \end{align*} assuming that $T_j$ have bounded imaginary powers, that is, for some $N_j\ge 1$ and $\theta_j \ge 0,$ $$ \big\|T_j^{is}\big\|_{\cB(\cH)} \leq N_j e^{\theta_j |s|}, \quad s \in \bbR, \; j=1,2. $$ We also derive analogous bounds with $\cB(\cH_1,\cH_2)$ replaced by trace ideals, $\cB_p(\cH_1, \cH_2)$, $p \in [1,\infty)$. The methods employed are elementary, predominantly relying on Hadamard's three-lines theorem and Heinz's inequality.