On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

Models for n-tuples of noncommuting operators

This paper proves versions of the Rota model theorem, the de Branges-Rovnyak model theorem, and the coisometric extension theorem for n-tuples of not necessarily commuting operators. This generalizes the work of A. E. Frazho (J. Funct. Anal.48 (1982), 1–11) for pairs of operators. The methods involve applying the single operator results to matrices of operators.     

On spectra of some linear elliptic differential operators

In the study of boundary-value problems, we often need information on the spectra of the corresponding differential operators. In this paper, we discuss the location of spectra of linear elliptic differential operators in ℝⁿ, n ∈ ℕ.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

On the norms of quaternionic harmonic projection operators

As a consequence of integral bounds for three classes of quaternionic spherical harmonics, we prove some bounds from below for the $(L^p,L^2)$ norm of quaternionic harmonic projectors, for $p\in [1,2]$.     

On independent varieties and some related notions

We investigate the relation of independence between varieties, as well as a generalisation of such which we call strict quasi-independence. Concerning the former notion, we specify a procedure for constructing an independent companion of a given solvable subvariety of a congruence modular variety; we show that joins of independent varieties inherit Mal’cev properties from the joinands; we investigate independence in 3- and 4-permutable varieties; we provide a more economical axiomatisation for the join of two independent varieties than the ones available in the literature. We also explore the latter notion, showing inter alia that joins of strictly quasi-independent varieties inherit the congruence extension property and the strong amalgamation property from the joinands, and conversely. An application section investigates independent varieties of Boolean algebras with operators (in particular, Akishev and Goldblatt’s bounded monadic algebras) and of groups. In particular, a complete characterisation of independent varieties of groups is given.     

On spectra ofp-hyponormal operators

The purpose of this paper is to show the following: Let 0<p<1/2. IfT=U|T| is a p-hyponormal operator with a unitaryU on a Hilbert space, then $$\sigma (T) = \mathop \cup \limits_{0 \leqslant k \leqslant 1} \sigma (T_{\left[ k \right]} ),$$ where $$T_{\left[ k \right]} = U[(1 - k)S_U^ - (\left| T \right|^{2p} ) + kS_U^ + (\left| T \right|^{2p} ]^{\tfrac{1}{{2p}}} $$ andS _U ^± (T) denote the polar symbols ofT.     

On a characterization of the essential spectra of some matrix operators and application to two-group transport operators

In this paper, we investigate the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to two-group transport operators in the Banach space L _p ([−a, a] × [−1, 1]) × L _p ([−a, a] × [−1, 1]), a > 0, p ≥ 1.     

Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S‐spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.     

On some notions of good reduction for endomorphisms of the projective line

Let Φ be an endomorphism of ${\mathbb{P}^1_{\overline{\mathbb{Q}}}}$ , the projective line over the algebraic closure of ${\mathbb{Q}}$ , of degree ≥ 2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds also for any pair of branch points. We say that Φ has simple good reduction at v if the map Φ _v , the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φ _v is separable, then Φ has simple good reduction at v.     

On the spectral geometry for the Jacobi operators of harmonic maps into a quaternionic projective space

We characterize both invariant and totally real immersions into the quaternionic projective space by the spectra of the Jacobi operator. Also, we study spectral characterization of harmonic submersions when the target manifold is the quaternionic projective space.     

On some operators inc 0

Theorem. Let S be a bounded Suslin set in the plane. Then there is a bounded linear operator T in c_o, whose point spectrum σ _e (T)=S.     

On some univalent integral operators

In this paper some new subclasses of Bazilevic functions are defined and it is shown that these classes are preserved under certain integral operators.