Some Operator Bounds Employing Complex Interpolation Revisited

Abstract

We revisit and extend known bounds on operator-valued functions of the type

$$T^{-Z}_{1}ST^{-1+z}_{2},\;z\in\bar{\Sigma}\;=\;\left\{z\in\mathbb{C}|\mathrm{Re}{z}\in[0,1]\right\}$$

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 $\mathcal{H}_{j},j\;=\;1,2,$ and suppose that S (resp., S*) is a densely defined closed operator mapping dom $(S)\subseteq \mathcal{H}_{1}\; \mathrm{into}\;\mathcal{H}_{2}\;(\mathrm{resp.}, \mathrm{dom}(S^{*})\subseteq \mathcal{H}_{2}\;\mathrm{into}\;\mathcal{H}_{1}$, relatively bounded with respect to $T_{1}\;(\mathrm{resp.,}\;T^{*}_{2})$. Using complex interpolation methods, a generalized polar decomposition for S, and (a variant of) the Loewner–Heinz inequality, the bounds we establish lead to inequalities of the following type: Given $k\;\in\;(0,\infty)$,

$$\|\overline{T^{-Z}_{1}ST^{-1+z}_{2}}\|_{\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})}\;\leq\;N_{1}N{2}e^{k(\mathrm{Im}(z))^2+k\; \mathrm{Re}(z)[1-\mathrm{Re}(z)]+(4k)^{-1}(\theta_{1}+\theta_{2})^2} \times\|ST^{-1}_{1}\|^{1-{\mathrm{Re}}(z)}_{\mathcal{B}(\mathcal{H_{1},\mathcal{H}_{2}})}\|S^{*}(T^{*}_{2})^{-1}\|^{{\mathrm{Re}}(z)}_{\mathcal{B}(\mathcal{H_{2},\mathcal{H}_{1}})},\qquad z\in\bar{\bar{\Sigma}}$$

which also implies,

$$\|\overline{T^{-Z}_{1}ST^{-1+z}_{2}}\|_{\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})}\;\leq\;N_{1}N{2}e^{(\theta_{1}+\theta_{2})[x(1-x)]^{1/2}} \times\|ST^{-1}_{1}\|^{1-{x}_{\mathcal{B}(\mathcal{H_{1},\mathcal{H}_{2}})}\|S^{*}(T^{*}_{2})^{-1}\|^{x}_{\mathcal{B}(\mathcal{H_{2},\mathcal{H}_{1}})},\qquad x\in[0,1]}$$

assuming that T _j have bounded imaginary powers, that is, for some $N_{j}\geqslant\;1\mathrm{and}\;\theta\;\geqslant\;0$,

$$\|T^{is}_{j}\|_\mathcal{B(H)}\;\leq\;N_{j}e^{\theta_{j}|s|},\quad s\in\mathbb{R}, j\;=\;1,2$$

. We also derive analogous bounds with $\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})$ replaced by trace ideals, $\mathcal{B}_{p}(\mathcal{H}_{1},\mathcal{H}_{2}),\;p\;\in\;[1,\infty)$. The methods employed are elementary, predominantly relying on Hadamard’s three-lines theorem and the Loewner–Heinz inequality.

Mathematics Subject Classification (2010). Primary 47A57, 47B10, 47B44; Secondary 47A30, 47B25