Variational principles are very powerful tools when studying self-adjoint linear operators on a Hilbert spaceH. Bounds for eigenvalues, comparison theorems, interlacing results and monotonicity of eigenvalues can be proved easily with these characterizations, to name just a few. In this paper we consider generalization of these principles to families of linear, self-adjoint operators depending ...

The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric opera...

The paper deals with difference schemes for the heat-conduction equation with nonlocal boundary conditions containing two real parameters, α and γ. Such schemes have been investigated for some special parameter values, but the general case was not considered previously. The eigenvalue problem arises as a result of variable division and is solved here explicitly. The so-called reality domains we...

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropy for finite sets of self-adjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebrai...

We investigate composition-differentiation operators acting on the Dirichlet space of unit disk. Specifically, we determine characterizations for bounded, compact, and Hilbert-Schmidt operators. In addition, particular classes inducing maps, derive an adjoint formula, compute norm, identify spectrum.

Let λ ∈ (0, 1) and let T be a r × r complex matrix with polar decomposition T = U |T |. Then, the λAluthge transform is defined by ∆λ (T ) = |T | U |T |. Let ∆nλ(T ) denote the n-times iterated Aluthge transform of T , n ∈ N. We prove that the sequence {∆nλ(T )}n∈N converges for every r × r diagonalizable matrix T . We show regularity results for the two parameter map (λ, T ) 7→ ∆∞λ (T ), and w...

We consider the two-dimensional Dirac operator with Lorentz-scalar \(\delta \)-shell interactions on each edge of a star graph. An orthogonal decomposition is performed which shows such an unitarily equivalent to sum half-line operators off-diagonal Coulomb potentials. This reduces computation deficiency indices determining number eigenvalues one-dimensional spin–orbit in interval \((-1/2,1/2)\...