Properties of some norm related functions of unbounded linear operators
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 Theorem 3.10). If dim D(T)\(=\infty\) then \(T\in F_ +\) \(\Leftrightarrow\) \(\Gamma (T)>0\) \(\Leftrightarrow\) \(\tau _ 0(T)>0\) \(\Leftrightarrow\) \({\bar \tau}{}_ 0(T)>0\) (Theorem 2.2). It follows that the perturbation class of \(F_ +\) is the class of strictly singular operators (Theorem 3.6). z(\({\bar \Gamma}{}_ 0)\) is the class of precompact operators (Theorem 3.2) and \(z(\Gamma _ 0)\) is the class of operators with a finite codimensional restriction which is precompact (Corollary 3.3). The class s(\({\bar \Gamma}{}_ 0)\) is the class of discontinuous operators while \(s(\Gamma _ 0)=s(\Delta)=s(\tau)\) is the class of operators which are discontinuous on every finite codimensional subspace. s(\(\tau)\) is by definition the class of operators whose only continuous restrictions are the finite dimensional ones, and \(s(\tau _ 0)\) consists of the operators having an infinite dimensional finite codimensional restriction with a precompact inverse (Theorem 4.8). A measure of noncompactness studied by \textit{A. Lebow} and \textit{M. Schechter} [J. Funct. Anal. 7, 1-26 (1971; Zbl 0209.450)], is investigated in the present context of unbounded operators: for a given subset \(\Omega\) of X define \(m(\Omega)=\inf \{r:\) \(\Omega\) is covered by finitely many open balls of radius \(r\}\). Theorem 5.10: \(T\in F_ +\) \(\Leftrightarrow\) there exists \(c>0\) such that m(\(\Omega\) \(\cap D(T))\leq cm(T\Omega)\) for all bounded subsets \(\Omega\) of X.
finite codimensional restriction having a continuous inverse
scientific article; zbMATH DE number 4042608
scientific article