We consider trace functions (A,B) 7→ Tr[(ABA)] where A and B are positive n × n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of ABA and convexity/concavity of the closely related trace functional Tr[ABAC]. The concavity questions are completely resolved, thereby settling cases left open by Hiai; the convexity questions are settled in...

This is based on joint work with R. T. Seeley. The introduction presents the problem of parameter-dependent calculi for do's and the question of trace asymptotics for Atiyah-Patodi-Singer operators. Chapter 2 establishes relations between the three operator functions: resolvent, heat operator and power operator (zeta function). Chapter 3 explains our parameter-dependent do calculus with weak po...

This article investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., “How many traces can an ideal support?” We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced in [13] and the analogous arithmetic mean operations at infinity (am-∞) that we develop extensi...

Let A(p) denote the class of functions which are analytic in the open unit disk U. By making use of certain integral operator,we obtain some interesting properties of multivalent analytic functions.

let a(p) denote the class of functions which are analytic in the open unit disk u. by making use of certain integral operator,we obtain some interesting properties of multivalent analytic functions.

We prove variants of Wiener's Tauberian theorem in the framework quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or two operators. Our results include as special case. Applications our theorems are related to localization operators, Toeplitz isomorphism Bargmann-Fock spaces quantization schemes with consequences Shubin's pse...

This paper concerns a rather concrete phenomenon in abstract operator algebras. The main examples of the algebras we study are algebras of singular integral operators (pseudo-differential operators of order zero). As everyone knows, the Fredhohn index of a pseudo-differential operator depends only on its symbol and the Atiyah-Singer Index Theorem gives an explicit formula for computing the depe...