On the functions of one selfadjoint integral operator

We construct a selfadjoint Akhiezer integral operator S on L ₂(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator.     

〈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.     

On Causal Invertibility with Respect to a Cone of Integral-Difference Operators in Vector Function Spaces

Let \(\mathbb{S}\) be a cone in ? ⁿ . A bounded linear operator T: L _p (? ⁿ ) → L _p (? ⁿ ) is said to be causal with respect to \(\mathbb{S}\) if the implication x(s) = 0 (s ε W ? \(\mathbb{S}\)) ? (Tx) (s) = 0 (s ε W ? \(\mathbb{S}\)) is valid for any x ε L _p (? ⁿ ) and any open subset W\(\subseteq\) ? ⁿ . The set of all causal operators is a Banach algebra. We describe the spectrum of the operator

$(Tx)(t) = \sum\limits_{n = 1}^\infty {a_n x(t - t_n )} + \int {\mathbb{S}g(s)x(t - s)ds,} \quad t \in \mathbb{R}^n ,$

in this algebra. Here x ranges in a Banach space \(\mathbb{E}\), the a _n are bounded linear operators in \(\mathbb{E}\), and the function g ranges in the set of bounded operators in \(\mathbb{E}\).

     

Mikhlin’s problem on Carnot groups

We consider one class of singular integral operators over the functions on domains of Carnot groups. We prove the L _p boundedness, 1 < p > ∞, for the operators of this class. Similar operators over the functions on domains of Euclidean space were considered by Mikhlin.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

Solvability of vector integro-differential equations of convolution type on the semiaxis

The paper deals with vector integro-differential equation of convolution type that have the form

$ - \frac{{d^2 f_i }}{{dx^2 }} + a_i f_i (x) = g_i (x) + \sum\limits_{j = 1}^N {\int\limits_0^\infty {K_{ij} (x - t)f_j (t)dt, } i = 1,2, \ldots ,N,} $

where \(\vec f\) = (f ₁, f ₂, ..., f _N ) ^T is the unknown vector-function, a _i are nonnegative numbers, \(\vec g\) = (g ₁, g ₂, ..., g _N ) ^T ? L ₁ ^×N (0,+∞) ≡ L ₁ (0,+∞) × ... × L (0,+∞) is the independent term of the equation with nonnegative components and 0 ≤ K _ij ? L ₁ (?∞,+∞), i, j = 1, 2, …,N are the kernel-functions. These equations have significant applications in the wave non-local interaction theory. Using some special factorization methods, solvability of the system is proved in different functional spaces.

     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the Schrödinger type operators

Let L _k = (?Δ) ^k + V ^k be a Schrödinger type operator, where k ≥ 1 is a positive integer and V is a nonnegative polynomial. We obtain the L ^p estimates for the operators ?^2k L _k ^?1 and ? ^k L _k ^?1/2 .     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Weighted estimates for commutators of singular integral operators with non-smooth kernels

In this paper, we establish the boundedness of commutators of singular integral operators with non-smooth kernels on weighted Lipschitz spaces Lipβ,ω. The condition on the kernel in this paper is weaker than the usual pointwise H¨ormander condition.     

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.     

Convolution Integral Operators

We establish some algebraic, metric, and spectral properties of convolution integral operators in L₂(?^N).     

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert <Emphasis Type="Italic">C</Emphasis>*-module <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">A</Emphasis>)

We characterize A-linear symmetric and contraction module operator semigroup{Tt}t∈R+L(l2(A)),where A is a finite-dimensional C-algebra,and L(l2(A))is the C-algebra of all adjointable module maps on l2(A).Next,we introduce the concept of operator-valued quadratic forms,and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l2(A)and the set of non-negative densely defined A-valued quadratic forms.In the end,we obtain that a real and strongly continuous symmetric semigroup{Tt}t∈R+L(l2(A))being Markovian if and only if the associated closed densely defined A-valued quadratic form is a Dirichlet form.     

Improved Hörmander’s theorem and new methods for oscillatory integral operators

This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.     

Equivalence of <Emphasis Type="Italic">K</Emphasis>-functionals and modulus of smoothness constructed by generalized Dunkl translations

In a Hilbert space L _2,α := L ₂(?, |x|^2α+1 dx), α > ? 1/2, we study the generalized Dunkl translations constructed by the Dunkl differential-difference operator. Using the generalized Dunkl translations, we define generalized modulus of smoothness in the space L _2,α . Based on the Dunkl operator we define Sobolev-type spaces and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness.     

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ^ρ with variable kernels on the Hardy spaces H ^p (R ⁿ ) and weak Hardy spaces WH ^p (R ⁿ ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H ^p,q(R ⁿ ) for all p < q < ∞.     

On estimates for the Fourier-Bessel integral transform in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(ℝ<Subscript>+</Subscript>)

Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L ₂(?₊) as applied to some classes of functions characterized by a generalized modulus of continuity.     

A Characterization of Stem Algebras in Terms of Central Derivations

Let L be a Lie algebra, and Der _z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der _z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der _z (L) is abelian if and only if L is a stem Lie algebra.     

Operators related to truncated Hilbert transforms on <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

In this paper, we give counterexamples to show that variation and oscillation operators related to rough truncations of the Hilbert transform are not bounded from H ¹ to L ¹, and prove variation, oscillation and λ-jump operators related to smooth truncations are bounded from H ¹ to L ¹.     

Traceability of Positive Integral Operators

A necessary and sufficient condition is given for a positive bounded linear operator with an integral kernel to be trace class on L²(μ) for a σ-finite measure μ. The condition refines earlier criteria for positive Hilbert–Schmidt operators and positive integral operators with continuous kernels on a locally compact space.