Abstract We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for trace class integral operator to be positive semidefinite, which is an important cornerstone quantum theory in phase-space representation. also present new, efficiently computable algorithm Newton’s identities. Our test positivity much more sensitive than the ones gi...

A classical result of Grothendieck and Lidskii says that the trace formula (that the trace of a nuclear operator is the sum of its eigenvalures provided the sequence of eigenvalues is absolutely summable) holds in Hilbert spaces. In 1988 Pisier proved that weak Hilbert spaces satisfy the trace formula. We exhibit a much larger class of Banach spaces, called Γ-spaces, that satisfy the trace form...

We classify functions $$f:(a,b)\rightarrow \mathbb {R}$$ which satisfy the inequality $$\begin{aligned} {\text {tr}}f(A)+f(C)\ge {tr}}f(B)+f(D) \end{aligned}$$ when $$A\le B\le C$$ are self-adjoint matrices, $$D= A+C-B$$ , so-called trace minmax functions. (Here B$$ if $$B-A$$ is positive semidefinite, and f evaluated via functional calculus.) A function only its derivative analytically continu...