An operator extension of the parallelogram law and related norm inequalities

Abstract

Abstract. We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let A be a C * -algebra, T be a locally compact Hausdorff space equipped with a Radon measure μ and let (A t ) t∈T be a continuous field of operators in A such that the function t → A t is norm continuous on T and the function t → A t is integrable. If α : T × T → C is a measurable function such that α(t,s)α(s,t) = 1 for all t,s ∈ T , then we show thatMathematics subject classification (2010): Primary 47A63; Secondary 46C15, 47A30, 47B10, 47B15, 15A60.

Some equivalent characterizations of inner product spaces and their consequences

“…Another version of "the generalized parallelogram law" could be considered the Corollary 2.7. from [11]. In this paper we will give another proof for this result.…”

“…As a consequence of Proposition 4.4., we obtain a very short proof for the next corollary. This result represents another characterization of the inner product spaces and it is known as a version of "the generalized parallelogram law" (see [11], pag. 3).…”

Some complementary inequalities to Jensen’s operator inequality

“…In [30], some techniques are used while manipulating some inequalities related to continuous fields of operators.…”

Schatten p-norm inequalities related to an extended operator parallelogram law

“…Generalizations of the parallelogram law for the Schatten p-norms have been given in the form of the celebrated Clarkson inequalities (see [4] and references therein). Since C 2 is a Hilbert space under the inner product A, B = tr(B * A), it follows from an equality similar to (1.2) stated for vectors of a Hilbert space (see [7,Corollary 2.7]) that if A 1 , . .…”

“…In [7] a joint operator extension of the Bohr and parallelogram inequalities is presented. In particular, it follows from [7, Corollary 2.3] that if A 1 , .…”