Some inequalities of the Slater type for convex functions of selfad-joint operators in Hilbert spaces H under suitable assumptions for the involved operators are given. Amongst others, it is shown that if A is a positive definite operator with Sp (A) ⊂ [m, M ] and f is convex and has a continuous derivative on [m, M ] , then for any x ∈ H with x = 1 the following inequality holds: 0 ≤ f Af ′ (A...

and Applied Analysis 3 where ‖ · ‖p p ∈ 1,∞ are the Lebesgue norms, that is, ∥∥f ′∥∞ ess sup s∈ a,b ∣∣f ′ s ∣∣, ∥∥f ′∥∥ p : (∫b a ∣∣f ′ s ∣∣ds)1/p, p ≥ 1. 1.6 The case of convex functions is as follows 4 . Theorem 1.5. Let f : a, b → be a convex function on a, b . Then one has the inequalities 1 8 b − a 2 [ f ′ ( a b 2 ) − f ′ − ( a b 2 )] ≤ f a f b 2 b − a − ∫b

We study homological structure of the filtration of the space of selfadjoint operators by the multiplicity of the ground state. We consider only operators acting in a finite dimensional complex or real Hilbert space but infinite dimensional generalizations are easily guessed.

In this paper we treat the case of an abstract vibrational system of the form Mẍ + Cẋ + x = 0, where the positive semi-definite selfadjoint operators M and C commute. We explicitly calculate the solution of the corresponding Lyapunov equation which enables us to obtain the set of optimal damping operators, thus extending already known results in the matrix case.

We introduce and study the variational problem to maximize the determinant of a random selfadjoint operators obtained from discrete abelian or nonabelian lattice gauge fields. We prove the existence of minima and give rough estimates of the functional for multi-particle operators.

The multivariate orthogonal polynomials are related to a family of commuting selfadjoint operators. The spectral theorem for these operators is used to prove that a polynomial sequence satisfying a vector-matrix form of the three-term relation is orthonormal with a determinate measure.

We consider the analytic continuation of the transfer function associated with a 2×2 operator matrix having unbounded couplings into unphysical sheets of its Riemann surface. We construct a family of non-selfadjoint operators which factorize the transfer function and reproduce certain parts of its spectrum including the nonreal (resonance) spectrum situated in the unphysical sheets neighboring ...

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schrödinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of ...

In this paper we start with the development of a theory of presheaves on a lattice, in particular on the quantum lattice L(H) of closed subspaces of a complex Hilbert space H, and their associated etale spaces. Even in this early state the theory has interesting applications to the theory of operator algebras and the foundations of quantum mechanics. Among other things we can show that classica...