Continuous and holomorphic functions with values in closed operators

TL;DR

This paper develops a systematic framework for analyzing continuous and holomorphic functions with values in closed operators, including criteria and properties, especially for operators with complex spectral characteristics.

Contribution

It introduces general criteria for continuity and holomorphicity of operator-valued functions, including sum and product operations, and employs graph-based methods to analyze operator functions.

Findings

Criteria for holomorphicity of operator-valued functions

Conditions for sum and product of holomorphic operator functions

Use of projections onto subspaces to analyze operator functions

Abstract

We systematically derive general properties of continuous and holomorphic functions with values in closed operators, allowing in particular for operators with empty resolvent set. We provide criteria for a given operator-valued function to be continuous or holomorphic. This includes sufficient conditions for the sum and product of operator-valued holomorphic functions to be holomorphic. Using graphs of operators, operator-valued functions are identified with functions with values in subspaces of a Banach space. A special role is thus played by projections onto closed subspaces of a Banach space, which depend holomorphically on a parameter.