Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions

We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in $\mathbb{R}^{N}$ . We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator P _s,t , which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any t>s, the evolution operator P _s,t is compact in the previous weighted spaces.     

The Dirac Operator of a Commuting d-Tuple

Given a commuting d-tuple T=(T₁, …, T_d) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator D_T. Significant attributes of the d-tuple are best expressed in terms of D_T, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, …) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d -contractions of finite rank. It is shown that for the subcategory of all such T that are (a) Fredholm and and (b) graded, the curvature invariant K(T) is stable under compact perturbations. We do not know if this stability persists when T is Fredholm but ungraded, although there is concrete evidence that it does.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

TWO-DIMENSIONAL MAXIMAL OPERATOR OF DYADIC DERIVATIVE ON VILENKIN MARTINGALE SPACES

In [1] the boundedness of one dimensional maximal operator of dyadic derivative is discussed.In this paper,we consider the two-dimensional maximal operator of dyadic derivative on Vilenkin martingale spaces.With the help of counter-example we prove that the maximal operator is not bounded from the Hardy space H q to the Hardy space H q for 0相似文献   

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

Multiplication Operators on Invariant Subspaces of Function Spaces

Let M_φ be the operator of multiplication by φ on a Hilbert space of functions analytic on the open unit disk. For an invariant subspace F for the multiplication operator M_z, we derive some spectral properties of the multiplication operator M_φ : F → F. We characterize norm, spectrum, essential norm and essential spectrum of such operators when F has the codimension n property with n ∈ {1, 2, …, + ∞}.     

Asymptotic formulas for fundamental solutions of the Dirac system with complex-valued integrable potential

We study the Dirac operator with a complex-valued integrable potential in the space ? = L ₂[0, π]⊕L ₂[0, π]. We obtain asymptotic formulas for a fundamental solution system of an operator. Remainders in each of the formulas are estimated.     

Generalization of Berg-Dimovski convolution in spaces of analytic functions

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator J _π+L where J _ρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator J _π+L in ?(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.     

On extensions of the Bessel operator on a finite interval and a half-line

The minimum and maximum operators generated by a Bessel differential expression are studied. The boundary triplets for the maximum operator A _??,max are constructed, and the corresponding Weyl functions are found. All non-negative self-adjoint extensions of the minimum operator A _??,min are described.     

Bifurcation for variational problems when the linearisation has no eigenvalues

For a bounded analytic function, ?, on the unit disk, $\text{D,}\text{let}\text{T}{}_{\mathrm{?}}\text{and}\text{M}{}_{\mathrm{?}}$ denote the operators of multiplication by ? on H²(?D) and L²(?D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator $\text{T}{}_{\mathrm{?}}$ that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L². In the first part of this paper, we give an explicit example of an analytic Toeplitz operator T_φ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of T_φ that extend to commute with M_φ are analytic Toeplitz operators. Although the commutant of T_φ contains more than just analytic Toeplitz operators, T_φ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether T_g commutes with nonzero compact operators, whether T_g is irreducible, and which operators in the commutant of T_g extend to the commutant of M_g. In particular, we find representations for operators in the commutant and second commutant of T_g.     

A class of Banach spaces with few non-strictly singular operators

We construct a family (X_γ) of reflexive Banach spaces with long (countable as well as uncountable) transfinite bases but with no unconditional basic sequences. The method we introduce to achieve this allows us to considerably control the structure of subspaces of the resulting spaces as well as to precisely describe the corresponding spaces on non-strictly singular operators. For example, for every pair of countable ordinals γ,β, we are able to decompose every bounded linear operator from X_γ to X_β as the sum of a diagonal operator and an strictly singular operator. We also show that every finite-dimensional subspace of any member X_γ of our class can be moved by and (4+?)-isomorphism to essentially any region of any other member X_δ or our class. Finally, we find subspaces X of X_γ such that the operator space L(X,X_γ) is quite rich but any bounded operator T from X into X is a strictly singular pertubation of a scalar multiple of the identity.     

Asymptotic analysis of non-self-adjoint Hill operators

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L _t (q) with a potential q ∈ L ₁[0,1] and t-periodic boundary conditions, t ∈ (?π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L ₂(?∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.     

Real hypersurfaces in a complex space form with a condition on the structure Jacobi operator

In this paper we classify the real hypersurfaces in a non-flat complex space form with its structure Jacobi operator R _ξ satisfying (? _X R _ξ )ξ = 0, for all vector fields X in the maximal holomorphic distribution D. With this result, we prove the non-existence of real hypersurfaces with D-parallel as well as D-recurrent structure Jacobi operator in complex projective and hyperbolic spaces. We can also prove the non-existence of real hypersurfaces with recurrent structure Jacobi operator in a non-flat complex space form as a corollary.     

Isometric properties of elementary operators

We consider the elementary operator L, acting on the Hilbert-Schmidt Class C₂(H), given by L(T)=ATB, with A and B bounded operators on H. We establish necessary and sufficient conditions on A and B for L to be a 2-isometry or a 3-isometry. We derive sufficient conditions for L to be an n-isometry. We also give several illustrative examples involving the weighted shift operator on l₂ and the multiplication operator on the Dirichlet space.     

Representations for Moore-Penrose inverses in Hilbert spaces

Let H₁, H₂ be two Hilbert spaces, and let T : H₁ → H₂ be a bounded linear operator with closed range. We present some representations of the perturbation for the Moore-Penrose inverse in Hilbert spaces for the case that the perturbation does not change the range or the null space of the operator.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type (A) in complex two plane Grassmannians G ₂(? ^m+2) with a commuting condition between the shape operator A and the structure tensors φ and φ ₁ for M in G ₂(? ^m+2). Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator A and a new operator φφ ₁ induced by two structure tensors φ and φ ₁. That is, this commuting shape operator is given by φφ ₁ A = A φφ ₁. Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G ₂(? ^m+1) in G ₂(? ^m+2).     

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A₀ acting on a Hilbert space H singular perturbations of the form A₀+V, are studied under some additional requirements of symmetry imposed on the initial operator A₀ and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A₀+V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L₂(R³) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.     

Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, Σ _dc spaces, where an infinite dimensional Banach space X is called a Σ _dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T ₁ and T ₂ on a Σ _dc space are similar if and only if the K ₀-group of the commutant algebra of the direct sum T ₁⊕T ₂ is isomorphic to the group of integers ?. On a Σ _dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in (ΣSI)(X), it further gives a necessary and sufficient condition that two operators in (ΣSI)(X) are similar by using the ordered K ₀-groups. It also proves that every operator in (ΣSI)(X) has a unique (SI) decomposition up to similarity on a Σ _dc space X, where (ΣSI)(X) denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.     

The best <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Close two-sided estimates are obtained for the best approximation in the space L _p (? ^m ), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L _p (? ^m ). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best.     

Functional differential equations of delay type and nonlinear evolution in Lp-spaces

Nonlinear nonautonomous functional differential equations of delay type with L_p-initial functions is studied by means of the theory of evolution operators in L_p-spaces. Using the evolution operator in C it is shown that the evolution operator constructed in L_p × Rⁿ determines the solutions of the initial value problem in the exact sense. A variation of parameters formula is developed for the nonlinear perturbed equation.