Abstract
We revisit and extend known bounds on operator-valued functions of the type
under various hypotheses on the linear operators S and Tj , j = 1, 2. We particularly single out the case of self-adjoint and sectorial operators Tj 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 \(k\;\in\;(0,\infty)\),
which also implies,
assuming that T j have bounded imaginary powers, that is, for some \(N_{j}\geqslant\;1\mathrm{and}\;\theta\;\geqslant\;0\),
. 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
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© 2015 Springer International Publishing Switzerland
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Gesztesy, F., Latushkin, Y., Sukochev, F., Tomilov, Y. (2015). Some Operator Bounds Employing Complex Interpolation Revisited. In: Arendt, W., Chill, R., Tomilov, Y. (eds) Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics. Operator Theory: Advances and Applications, vol 250. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18494-4_14
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DOI: https://doi.org/10.1007/978-3-319-18494-4_14
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-18493-7
Online ISBN: 978-3-319-18494-4
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