In this paper we prove sharp estimates for generalized eigenvectors of Hermitian Jacobi matrices associated with the spectral parameter lying in a gap of their essential spectra. The estimates do not depend on the main diagonals of these matrices. The types of estimates obtained for bounded and unbounded gaps are different. These estimates extend the previous ones found in [J. Janas, S. Naboko and G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not.4 (2009), 736–764], for the spectral parameter being outside the whole spectrum of Jacobi matrices. Examples illustrating optimality of our results are also given.

We consider uniformly elliptic operators with Dirichlet or Neumann homogeneous boundary conditions on a domain Ω in ℝN. We consider deformations ϕ(Ω) of Ω obtained by means of a locally Lipschitz homeomorphism ϕ and we estimate the variation of the eigenfunctions and eigenvalues upon variation of ϕ. We prove general stability estimates without assuming uniform upper bounds for the gradients of the maps ϕ. As an application, we obtain estimates on the rate of convergence for eigenvalues and eigenfunctions when a domain with an outward cusp is approximated by a sequence of Lipschitz domains.

In general, multiplication of operators is not essentially commutative in an algebra generated by integral-type operators and composition operators. In this paper, we characterize the essential commutativity of the integral operators and composition operators from a mixed-norm space to a Bloch-type space, and give a complete description of the universal set of integral operators. Corresponding results for boundedness and compactness are also obtained.

Let 𝒟 be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space X and let δ:Alg 𝒟→Alg 𝒟 be a linear mapping. We show that δ is Jordan derivable at zero, that is, δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB+BA=0 if and only if δ has the form δ(A)=τ(A)+λA for some derivation τ and some scalar λ. We also show that if the dimension of X is greater than 2, then δ satisfies δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A)whenever AB=0if and only if δ is a derivation.

We characterize straightness of digital curves in the integer plane by means of difference operators. Earlier definitions of digital rectilinear segments have used, respectively, Rosenfeld’s chord property, word combinatorics, Reveillès’ double Diophantine inequalities, and the author’s refined hyperplanes. We prove that all these definitions are equivalent. We also characterize convexity of integer-valued functions on the integers with the help of difference operators.

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

We study Markov measures and p-adic random walks with the use of states on the Cuntz algebras Op. Via the Gelfand–Naimark–Segal construction, these come from families of representations of Op. We prove that these representations reflect selfsimilarity especially well. In this paper, we consider a Cuntz–Krieger type algebra where the adjacency matrix depends on a parameter q ( q=1 is the case of Cuntz–Krieger algebra). This is an ongoing work generalizing a construction of certain measures associated to random walks on graphs.

Let ℒ be a commutative subspace lattice and 𝒜=Alg ℒ. It is shown that every Jordan higher derivation from 𝒜 into itself is a higher derivation. We say that D=(δi)i∈ℕ is a higher derivable linear mapping at G if δn(AB)=∑ i+j=nδi(A)δj(B) for all n∈ℕ and A,B∈𝒜 with AB=G. We also prove that if D=(δi)i∈ℕ is a bounded higher derivable linear mapping at 0 from 𝒜 into itself and δn (I)=0for all n≥1 , or D=(δi)i∈ℕ is a higher derivable linear mapping at I from 𝒜 into itself, then D=(δi)i∈ℕ is a higher derivation.

We use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2.

Let A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

In this paper, we discuss the H1L-boundedness of commutators of Riesz transforms associated with the Schrödinger operator L=−△+V, where H1L(Rn) is the Hardy space associated with L. We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q>n/2. Let T1=V (x)(−△+V )−1, T2=V1/2(−△+V )−1/2 and T3 =∇(−△+V )−1/2. We prove that, for b∈BMO (Rn) , the commutator [b,T3 ]is not bounded from H1L(Rn)to L1 (Rn)as T3 itself. As an alternative, we obtain that [b,Ti ] , ( i=1,2,3 ) are of (H1L,L1weak) -boundedness.

Let A and B be C*-algebras, let X be an essential Banach A-bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T(a)T(b) + T(b)T(a) = 0 and S(a)b + bS(a) + aS(b) + S(b)a = 0 whenever a, b ε A are such that ab = ba = 0. We prove that then T = wΦ and S = D + Ψ, where w lies in the centre of the multiplier algebra of B, Φ: A → B is a Jordan epimorphism, D: A → X is a derivation and Ψ: A → X is a bimodule homomorphism.

We consider spectral radius algebras associated with C0 contractions. When the operator A is algebraic, we describe all invariant subspaces that are common for operators in its spectral radius algebra ℬA. When the operator A is not algebraic, ℬA is weakly dense and we characterize a set of rank-one operators in ℬA that is weakly dense in ℒ(ℋ).

Let L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra every nonzero homomorphism extends uniquely to a continuous homomorphism which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on (including the algebra of absolutely convergent Taylor series on ) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism which is also continuous with respect to the weak-star topologies.

A new notion of L(n)-hyponormality is introduced in order to provide a bridge between subnormality and paranormality, two concepts which have received considerable attention from operator theorists since the 1950s. Criteria for L(n)-hyponormality are given. Relationships to other notions of hyponormality are discussed in the context of weighted shift and composition operators.

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

We give derivative-free characterizations for bounded and compact generalized composition operators between (little) Zygmund type spaces. To obtain these results, we extend Pavlović’s corresponding result for bounded composition operators between analytic Lipschitz spaces.

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic (‘double root’) situation. For the model with ‘non-smooth’ matrix entries we obtain the asymptotics of generalized eigenvectors and analyse the spectrum. In addition, we reformulate a very helpful theorem from a paper by Janas and Moszynski in its full generality in order to serve the needs of our method.

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie in the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-sided estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method and illustrate our results with various numerical experiments.

On a separable C*-algebra A every (completely) bounded map which preserves closed two-sided ideals can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at ∞ of locally trivial C*-bundles of finite type.