Greatest crossnorm

Abstract

Let .pbe a complex Hilbert space.If T is a completely continuous operator on .pthen (T*nl/2 is also completely continuous and nonnegative.If A" A2, . . .represe nt all the nonzero eigenvalues of (T*n' /2 -each eigenvalue repeated in the sequence the numbe r of times equal to its multiplicitywe may form the sum li Ai which we denote by T(n.By definition, the trace-class (TC) consists of all those operators T for which T(n is finite.(TC) forms a linear space and T(n de fin es there a norm.The resulting normed linear space turns out to be co mplete, and the operators of finite rank form a dense set in (TC).. It is of significance to observ e that for operators T of finit e rank , T(n may be also ex pressed via conce pts mea ningful in a perfec tly ge neral Banac h s pace .This observati on permits then to carry ove r to perfectly ge neral Banac h s paces the co ncept of a trace-c lass of operators: One co nsid e rs the linea r space of al l the ope rators T of finit e ran k on th e give n Banach space.There one defines T(n vi a th e co ncepts mea ningful in ge neral Banach spaces.Th e custom ary me t ric co mpl etion of the so res ultin g norm ed lin ear s pace furni shes th e n the desired trace-class of operato rs.