Elementary operators and orthogonality Original Research Article

Let image be a separable infinite dimensional Hilbert space and let image denote the algebra of operators on image into itself. We study the elementary operators image defined by φ(X)=∑i=1kAiXBi and φ*(X)=∑i=1kAi*XBi*. We prove that: (i) when short parallelφshort parallelless-than-or-equals, slant1, then short parallelφ(X)−X+Sshort parallelgreater-or-equal, slantedshort parallelSshort parallel for all image and all Sset membership, variantkerφ; (ii) when ∑i=1kAiAi*less-than-or-equals, slant1, ∑i=1kAi*Ailess-than-or-equals, slant1, ∑i=1kBiBi*less-than-or-equals, slant1 and ∑i=1kBi*Biless-than-or-equals, slant1, then for image (the von Neumann–Schatten p-class), 1less-than-or-equals, slantp<∞, short parallelφ(X)−X+Sshort parallelpgreater-or-equal, slantedshort parallelSshort parallelp and short parallelφ*(X)−X+Sshort parallelpgreater-or-equal, slantedshort parallelSshort parallelp for all image, where by convention short parallelYshort parallelp=∞ if image; (iii) let (Mi)i=1k and (Ni)i=1k be separately commuting sequences of normal operators and let image be defined by Δ(X)=∑i=1kMiXNi. If image and image, then short parallelΔ(X)+Sshort parallel22=short parallelΔ(X)short parallel22+short parallelSshort parallel22.