Clark model in the general situation

For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ?γ? = 1. Namely, all such unitary perturbations are operators U_γ:= U + (γ ? 1)( ·, b₁)_Hb, where b ∈ H, ∥b∥ = 1, b₁ = U^?1b, ?γ? = 1. For ?γ? < 1, the operators U_γ are contractions with one-dimensional defects.     

〈2, 1〉-Compact operators

We consider the class of the continuous L _2,1 linear operators in L ₂ that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L ₂ into a compact subset of L ₁. We prove that a functional equation with an operator from L _2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L _2,1 and VTV ^?1 ∈ L _2,1 for all unitary operators V in L ₂ then T = α1 + C, where α is a scalar, 1 is the identity operator in L ₂, and C is a compact operator in L ₂.     

To the Spectral Theory of One-Dimensional Matrix Dirac Operators with Point Matrix Interactions

We investigate one-dimensional (2p × 2p)-matrix Dirac operators D_X,α and D_X,β with point matrix interactions on a discrete set X. Several results of [4] are generalized to the case of (p × p)-matrix interactions with p > 1. It is shown that a number of properties of the operators D_X,α and D_X,β (self-adjointness, discreteness of the spectrum, etc.) are identical to the corresponding properties of some Jacobi matrices B_X,α and B_X,β with (p × p)-matrix entries. The relationship found is used to describe these properties as well as conditions of continuity and absolute continuity of the spectra of the operators D_X,α and D_X,β. Also the non-relativistic limit at the velocity of light c → ∞ is studied.     

A predual of a predual of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript> and its applications to commutators

A predual of B_σ-spaces is investigated. A predual of a predual of B_σ-spaces is also investigated, which can be used to investigate the boundedness property of the commutators. The relation between Herz spaces and local Morrey spaces is discussed. As an application of the duality results, one obtains the boundedness of the singular integral operators, the Hardy-Littlewood maximal operators and the fractional integral operators, as well as the commutators generated by the bounded mean oscillation (BMO) and the singular integral operators. What is new in this paper is that we do not have to depend on the specific structure of the operators. The results on the boundedness of operators are formulated in terms of ?_σ-spaces and B_σ-spaces together with the detailed comparison of the ones in Herz spaces and local Morrey spaces. Another application is the nonsmooth atomic decomposition adapted to B_σ-spaces.     

Commuting Krichever–Novikov differential operators with polynomial coefficients

Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form w² = z³+c₂z2+c₁z+c₀ with arbitrary coefficients c_i, there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree.     

Estimates of reachable sets of control systems with nonlinearity and parametric perturbations

In the Banach space L₁(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L₂(M, τ) is introduced, and its main properties are established. A convergence criterion in L₂(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L₁(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖₁ ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L₁(M, τ) is complemented. The convergence in L₂(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.     

Two classes of <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P ₁ and P ₂ of τ-measurable operators and investigate their properties. The class P ₂ contains P ₁. If a τ-measurable operator T is hyponormal, then T lies in P ₁; if an operator T lies in P _k , then UTU* belongs to P _k for all isometries U from M and k = 1, 2; if an operator T from P ₁ admits the bounded inverse T ^?1, then T ^?1 lies in P ₁. We establish some new inequalities for rearrangements of operators from P ₁. If a τ-measurable operator T is hyponormal and T ⁿ is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P ₁ coincides with the set of all paranormal operators on H.     

Differential operators for Siegel-Jacobi forms

For any positive integers n and m, H_(n,m):= H_n× C~(m,n) is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. We compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we construct a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for H_(n,m) are obtained.     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

Eigenfunctions of Composition Operators on Bloch-type Spaces

Suppose φ is a holomorphic self map of the unit disk and C_φ is a composition operator with symbol φ that fixes the origin and 0 < |φ'(0)| < 1. This paper explores sufficient conditions that ensure all the holomorphic solutions of Schröder equation for the composition operator C_φ to belong to a Bloch-type space B_α for some α > 0. In the second part of the paper, the results obtained for composition operators are extended to the case of weighted composition operators.     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

The essential spectrum of the three-particle discrete operator corresponding to a system of three fermions on a lattice

We consider a family of three-particle discrete Shrödinger operators H _μ (K). These operators are associated with the Hamiltonian for a system of three identical particles (fermions) with pairwise two-particle interactions on neighboring junctions of the d-dimensional lattice Z ^d . We describe the location and the structure of the essential spectrum of the operator H _μ (K) for all values of the three-particle quasi-momentum K ∈ T ^d and the interaction energy μ > 0.     

Positive operators on a free Banach space over the complex Levi-Civita field

Let C be the complex Levi-Civita field and let c ₀(C) or, simply, c ₀ denote the space of all null sequences of elements of C. A non-Archimedean norm is defined naturally on c ₀ with respect to which c ₀ is a Banach space. In this paper, we study the properties of positive operators on c ₀ which are similar to those of positive operators in classical functional analysis; however the proofs of many of the results are nonclassical. Then we use our study of positive operators to introduce a partial order on the set of compact and self-adjoint operators on c ₀ and study the properties of that partial order.     

Dynamical magnetic susceptibility in the spin-fermion model for cuprate superconductors

Using the method of diagram techniques for the spin and Fermi operators in the framework of the SU(2)-invariant spin-fermion model of the electron structure of the CuO₂plane of copper oxides, we obtain an exact representation of the Matsubara Green’s function D_⊥(k, iω _m ) of the subsystem of localized spins. This representation includes the Larkin mass operator Σ_L(k, iω _m ) and the strength and polarization operators P(k, iω _m ) and Π(k, iω _m ). The calculation in the one-loop approximation of the mass and strength operators for the Heisenberg spin system in the quantum spin-liquid state allows writing the Green’s function D_⊥(k, iω _m ) explicitly and establishing a relation to the result of Shimahara and Takada. An essential point in the developed approach is taking the spin-polaron nature of the Fermi quasiparticles in the spin-fermion model into account in finding the contribution of oxygen holes to the spin response in terms of the polarization operator Π(k, iω _m ).     

New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Let u be a holomorphic function and φ a holomorphic self-map of the open unit disk \(\mathbb{D}\) in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators uC _φ from Zygmund type spaces to Bloch type spaces in \(\mathbb{D}\) in terms of u, φ, their derivatives, and φ ⁿ , the n-th power of φ. Moreover, we obtain some similar estimates for the essential norms of the operators uC _φ , from which sufficient and necessary conditions of compactness of uC _φ follows immediately.     

Inner derivations of simple Lie pencils of rank 1

We prove that simple Lie pencils of rank 1 over an algebraically closed field P of characteristic 0 with operators of left multiplication being derivations are of the form of a sandwich algebra M ₃(U,D′), where U is the subspace of all skew-symmetric matrices in M ₃(P) and D′ is any subspace containing 〈E〉 in the space of all diagonal matrices D in M ₃(P).     

Extrapolation in Grand Lebesgue Spaces with <Emphasis Type="Italic">A</Emphasis>∞ Weights

Results on extrapolation withA∞ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the productmeasure μ₁ ×· · ·×μ_n onX₁ ×· · ·×X_n, where (X_i, d_i, μ_i), i = 1,..., n, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with A_∞ weights for operators of harmonic analysis are derived.     

Kolmogorov type inequalities for norms of Riesz derivatives of multivariate functions and some applications

Let L _∞,s ¹ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that ?f/?x _i ∈ L _s (? ^{m) for each i = 1, ...,m} . New sharp Kolmogorov type inequalities are obtained for the norms of the Riesz derivatives ∥D ^α f∥_∞ of functions f ∈ L _∞,s ¹ (? ^m ). Stechkin’s problem on approximation of unbounded operators D ^α by bounded operators on the class of functions f ∈ L _∞,s ¹ (? ^m ) such that ∥?f∥ _s ≤ 1 and the problem of optimal recovery of the operator D ^α on elements from this class given with error δ are solved.     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.