Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that $E={\mathbf{R}}^{m+1}$ is given the presheaf of solutions of$$\sum _{i=1}^m\frac{\partial u}{\partial x_i}=\frac{\partial u}{\partial x_{m+1}}$$and $B$ is a subset of ${\mathbf{R}}^m\times [-\infty ,0]$ satisfying$$\left\{\right(\alpha x,{\alpha }^{2}t):(x,t)\in B,x\in {\mathbf{R}}^m,t\in \mathbf{R}\}\subset B$$for $\alpha \>0$ arbitrarily small. Then $B$ is thin at 0 if and only if $B$ is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several approximation theorems on reducing and balayage...

We give a complete characterization of the positive trigonometric polynomials $Q(\theta ,\varphi )$ on the bi-circle, which can be factored as $Q(\theta ,\varphi )=|p(e^{i\theta },e^{i\varphi }){|}^2$ where $p(z,w)$ is a polynomial nonzero for $\left|z\right|=1$ and $\left|w\right|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4{\pi }^{2}Q(\theta ,\varphi )}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating...

A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak$\check{\mathrm{s}}$ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.

We completely characterize the ranks of A - B and $A^{1/2}-B^{1/2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m=rank(A^{1/2}-B^{1/2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of $rank(A^{1/2}-B^{1/2})$. For...

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is defined to be spectrally bounded if there is a constant M ≥ 0 such that r(Tx) ≤ Mr(x) for all x ∈ E, where r(·) denotes the spectral radius. We study some basic properties of this class of operators, which are sometimes analogous to, sometimes very different from, those of bounded operators between Banach spaces.

In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded $H^{\infty }$-calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function are not....

The formula $\varrho \left(M\right)=max{\varrho }_{\chi }\left(M\right),r\left(M\right)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), ${\varrho }_{\chi }\left(M\right)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.