On some inequalities with operators in Hilbert spaces

Abstract

Operators deï¬ned on Hilbert spaces represent a major subï¬eld of (or base for) the Functional Analysis. Several types of inequalities among such operators were established and studied in the last decades, mainly by the early ’50s and then in the ’80s. In this paper there are reviewed some of the most important types of inequalities, introduced and studied by H. Bohr, E. Heinz - T. Kato, H. Weyl, W. Reid and other authors. They were extended and/or sharpened by other authors, mentioned in the Introduction. Some of the deï¬nitions and proofs, found in several references, are completed (by the author) with speciï¬c formulas from Hilbert space theory, some details are also added to certain proofs and deï¬nitions as well. The main ways for establishing inequalities with operators are pointed out: scalar inequalities like the Cauchy-Schwarz and Bohr’s inequalities over C (the complex field) or on an H-space H, certain identities with H-space operators, etc.