Properties of some norm related functions of unbounded linear operators
type: Item
instance of: scholarly article
title: Properties of some norm related functions of unbounded linear operators
zbMATH Open document ID: 0639.47009
DOI: 10.1007/BF01159657
author: R. W. Cross
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173967
review text: Let L(X,Y) denote the set of linear transformations (``operators'') T: D(T)\(\subset X\to Y\) where X and Y are normed spaces. Let \({\mathcal I}(X)\), \({\mathcal C}(X)\) and \({\mathcal P}(X)\) denote the infinite dimensional, finite dimensional, and closed finite codimensional subspaces of X. Write \(S_ X=\{x\in X:\) \(\| x\| =1\}\). Let \({\mathcal A}\) be the set \(\{\Gamma,\Gamma _ 0,{\bar \Gamma}_ 0,\Delta,\tau,\tau _ 0,{\bar \tau}_ 0\}\) where each \(f\in {\mathcal A}\) is a function defined on every L(X,Y) as follows: if dim D(T)\(<\infty\) then \(f(T)=0\); otherwise \(\Gamma (T)=\inf \{\| T/M\|:\) \(M\in {\mathcal I}(D(T))\}\), \(\Gamma _ 0(T)=\inf \{\| T/M\|:\) \(T\in {\mathcal C}(x)\}\), \({\bar \Gamma}{}_ 0(T)=\inf \{\| T/M\|:\) \(T\in {\mathcal P}(x)\}\), \(\Delta (T)=\sup \{\Gamma (T/M):\) \(M\in {\mathcal I}(D(T))\}\), \(\tau (T)=\sup \{{\hat \gamma}_ M(T):\) \(M\in {\mathcal I}(D(T))\}\), \(\tau _ 0(T)=\sup \{{\hat \gamma}_ M(T):\) \(M\in {\mathcal C}(X)\}\), and \({\bar \tau}{}_ 0(T)=\sup \{{\hat \gamma}_ M(T):\) \(M\in {\mathcal P}(X)\}\) where \({\hat \gamma}{}_ M(T)=\inf \{\| Tx\|:\) \(x\in S_ M\}\). Let \(z(f)=\{T:\) \(f(T)=0\}\) and \(s(f)=\{T:\) \(f(T)=\infty \}\) (f\(\in {\mathcal A})\). The operator T is called an \(F_ +\)-operator if it has a finite codimensional restriction having a continuous inverse. When X and Y are complete and T is closed then \(T\in F_ +\) if and only if T is an upper semi-Fredholm operator. T is called strictly singular if \((T/M)^{-1}\) exists and is continuous only if dim \(M\cap D(T)<\infty\). In this paper the classes z(f) and s(f) are characterised (except for s(\({\bar \tau}{}_ 0))\). The functions \(\Delta\), \(\Gamma _ 0\) and \({\bar \Gamma}{}_ 0\) are subadditive and submultiplicative, while \(\tau _ 0\) is supermultiplicative (Propositions 2.12, 2.14 and 2.17, and Corollaries 2.8 and 2.10). T is strictly singular \(\Leftrightarrow\) \(\Delta (T)=0\) \(\Leftrightarrow\) \(\tau (T)=0\) (Proposition 3.4 and…
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 4042608
zbMATH Keywords: \(F_ +\)-operator
zbMATH Keywords: finite codimensional restriction having a continuous inverse
zbMATH Keywords: semi-Fredholm operator
zbMATH Keywords: strictly singular
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4042608