In the current work a generalization of the famous Weyl-Kodaira inversion formulas for the case of self-adjoint differential vector-operators is proved. A formula for spectral resolutions over an analytical defining set of solutions is discussed. The article is the first part of the planned two-part survey on the structural spectral theory of self-adjoint differential vector-operators in matrix Hilbert spaces.

We study asymptotic behavior of $C_0$-semigroups T(t), t ≥ 0, such that ∥T(t)∥ ≤ α(t), where α(t) is a nonquasianalytic weight function. In particular, we show that if σ(A) ∩ iℝ is countable and Pσ(A*) ∩ iℝ is empty, then $li{m}_{t\to \infty }1/\alpha \left(t\right)\parallel T\left(t\right)x\parallel =0$, ∀x ∈ X. If, moreover, f is a function in $L_{\alpha }^1\left({ℝ}_{+}\right)$ which is of spectral synthesis in a corresponding algebra $L_{{\alpha }_1}^1\left(ℝ\right)$ with respect to (iσ(A)) ∩ ℝ, then $li{m}_{t\to \infty }1/\alpha \left(t\right)\parallel T\left(t\right)f̂\left(T\right)\parallel =0$, where $f̂\left(T\right)={ʃ}_0^{\infty }f\left(t\right)T\left(t\right)dt$. Analogous results are obtained also for iterates of a single operator. The results are extensions of earlier results of Katznelson-Tzafriri,...

We consider simultaneous solutions of operator Sylvester equations $A_{i}X-X{B}_i=C_i$ (1 ≤ i ≤ k), where $(A₁,...,A_k)$ and $(B₁,...,B_k)$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C₁,...,C_k)$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A₁,...,A_k)$ and $(B₁,...,B_k)$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Let $T$ be an operator acting on a Banach space $X$, let $\sigma \left(T\right)$ and ${\sigma }_{BW}\left(T\right)$ be respectively the spectrum and the B-Weyl spectrum of $T$. We say that $T$ satisfies the generalized Weyl’s theorem if ${\sigma }_{BW}\left(T\right)=\sigma \left(T\right)\setminus E\left(T\right)$, where $E\left(T\right)$ is the set of all isolated eigenvalues of $T$. The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$, extending an earlier result of J. K. Finch and a...

The purpose of this paper is to give singular integral models for p-hyponormal operators and apply them to the Riemann-Hilbert problem.

We prove that the absolutely continuous part of the periodic Jacobi operator does not change (modulo unitary equivalence) under additive perturbations by compact Jacobi operators with weights and diagonals defined in terms of the Stolz classes of slowly oscillating sequences. This result substantially generalizes many previous results, e.g., the one which can be obtained directly by the abstract trace class perturbation theorem of Kato-Rosenblum. It also generalizes several results concerning perturbations...

Let X be a separable Banach space and denote by 𝓛(X) (resp. 𝒦(ℂ)) the set of all bounded linear operators on X (resp. the set of all compact subsets of ℂ). We show that the maps from 𝓛(X) into 𝒦(ℂ) which assign to each element of 𝓛(X) its spectrum, approximate point spectrum, essential spectrum, Weyl essential spectrum, Browder essential spectrum, respectively, are Borel maps, where 𝓛(X) (resp. 𝒦(ℂ)) is endowed with the strong operator topology (resp. Hausdorff topology). This enables us...