Entire functions and compact operators with \(S_p\)-imaginary component. (Q2782551)

Let \(S_p\) the von Neumann-Schatten class of compact operators in a Hilbert space \(H\). For any operator \(A\) of \(S_p\), the Hermitian parts are \(G=(A+A^*)/2\) and \(H=(A-A^*)/(2i)\). It is well known that if \(A=G+iH\) is a compact quasinilpotent operator, then \(\|A\| \leq2\|G\|=2\|H\|\). A weakly type of this relation has been given in [\textit{S. G. Kreĭn}, Sov. Math., Dokl. 1, 61--64 (1960); transl. from Dokl. Akad.Nauk SSSR 130, 491--494 (1960; Zbl 0089.32202)] and [\textit{V.I. Matsaev}, Sov. Math., Dokl. 2, 1013--1016 (1961); transl. from Dokl. Akad. Nauk SSSR 139, 810--813 (1961; Zbl 0119.32202)] for any compact quasinilpotent operator with imaginary part of traces class. The original proofs of these papers were based on the theory of entire functions. In this paper, the authors give a new approach based on entire functions and generalize, in the same way, the previous relations.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].