Some applications of Schwarz Lemma for operators

Mishra, Akshaya

doi:10.1155/s0161171289000402

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Introduction and Definitions1

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2016

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“…[4] to [9]; see also [1]) investigated various properties and characteristics of analytic functions of proper contractions, which include (among other results) operator versions of the Schwarz lemma, subordination theorems, Julia theorem, Pick-Julia theorem, Harnack's inequalities, Wolff's theorem, growth and distortion theorems, and so on. Mishra [14] also proved a sharpened form of the Schwarz lemma and Harnack's type inequalities for analytic functions of proper contractions. His results were motivated by the celebrated works of von Neumann [15], Potapov [16] and others.…”

Section: Introduction and Definitionsmentioning

confidence: 97%

Pick’s theorems for dissipative operators

Srivástava

Mishra

2016

Filomat

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Let H be a complex Hilbert space and let A be a bounded linear transformation on H. For a complex-valued function f , which is analytic in a domain D of the complex plane containing the spectrum of A, let f (A) denote the operator on H defined by means of the Riesz-Dunford integral. In the present paper, several (presumably new) versions of Pick's theorems are proved for f (A), where A is a dissipative operator (or a proper contraction) and f is a suitable analytic function in the domain D.

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