Operator Topologies

Abstract

107. Topologies for operators. A Hilbert space has two useful topologies (weak and strong); the space of operators on a Hilbert space has several. The metric topology induced by the norm is one of them; to distinguish it from the others, it is usually called the norm topology or the uniform topology. The next two are natural outgrowths for operators of the strong and weak topologies for vectors. A subbase for the strong operator topology is the collection of all sets of the form $$\left\{ {A:\,\parallel \,\left( {A\, - \,{A_0}} \right)f\parallel \, < \,\varepsilon } \right\};$$ correspondingly a base is the collection of all sets of the form $$\left\{ {A:\,\,\parallel \,\left( {A\, - \,{A_0}} \right){f_i}\,\parallel \,\, < \,\varepsilon ,\,\,i\, = \,1,\,...\,,\,k} \right\}.$$. Here k is a positive integer, f 1 … , f k are vectors, and ε is a positive number. A subbase for the weak operator topology is the collection of all sets of the form $$\left\{ {A:\left| {\left( {\left( {A\, - \,{A_0}} \right)\,f,\,g} \right)} \right|\, < \,\varepsilon } \right\},$$ where f and g are vectors and ε > 0; as above (as always) a base is the collection of all finite intersections of such sets. The corresponding concepts of convergence (for sequences and nets) are easy to describe: A n → A strongly if and only if A n f → Af strongly for each f (i.e., ||(A n – A)f|| → 0 for each f), and A n → A weakly if and only if A n f → Af weakly for each f(i.e.,(A n f, g) → (Af, g) for each f and g). For a slightly different and often very efficient definition of the strong and weak operator topologies see Problems 224 and 225.KeywordsHilbert SpaceWeak TopologyNorm TopologySequential ContinuityStrong TopologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.