Some operator bounds employing complex interpolation revisited

Abstract

We revisit and extend known bounds on operator-valued functions of the type T−z1 ST−1+z2, z∈ Σ = {z ∈ ℂ| Re(z) ∈ 0, 1]}, under various hypotheses on the linear operators S and Tj , j = 1, 2. We particularly single out the case of self-adjoint and sectorial operators j in some separable complex Hilbert space j , j = 1, 2, and suppose that S (resp., S∗) is a densely defined closed operator mapping dom(S) ⊆ H1 into H2 (resp., dom(S∗) ⊆ H2 into H1), relatively bounded with respect to T1 (resp., T∗2). Using complex interpolation methods, a generalized polar decomposition for S, and (a variant of) the Loewner–Heinz inequality, the bounds we establish lead to inequalities of the following type: Given k ∈ (0,∞), (Formula Presented), assuming that Tj have bounded imaginary powers, that is, for (Formula Presented). We also derive analogous bounds with B(H1,H2) replaced by trace ideals, Bp(H1,H2), p ∈ 1,∞). The methods employed are elementary, predominantly relying on Hadamard’s three-lines theorem and the Loewner–Heinz inequality.