Showing papers in "Integral Equations and Operator Theory in 2011"

General Fractional Calculus, Evolution Equations, and Renewal Processes

Abstract: We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form \({(\mathbb D_{(k)} u)(t)=\frac{d}{dt} \int olimits_0^tk(t-\tau )u(\tau )\,d\tau-k(t)u(0)}\) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation \({\mathbb D_{(k)} u=-\lambda u}\), λ > 0, proved to be (under some conditions upon k) continuous on [0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Abstract: In this paper the authors study the boundedness for a large class of sublinear operators \({T_{\alpha}, \alpha \in [0,n)}\) generated by Calderon–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces \({M_{p,\varphi}}\) . As an application of the above result, the boundeness of the commutator of sublinear operators \({T_{b,\alpha}, \alpha \in [0,n)}\) on generalized Morrey spaces is also obtained. In the case \({b \in BMO}\) and Tb,α is a sublinear operator, we find the sufficient conditions on the pair \({(\varphi_1,\varphi_2)}\) which ensures the boundedness of the operators \({T_{b,\alpha}, \alpha \in [0,n)}\) from one generalized Morrey space \({M_{p,\varphi_1}}\) to another \({M_{q,\varphi_2}}\) with 1/p − 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({(\varphi_1,\varphi_2)}\) , which do not assume any assumption on monotonicity of \({\varphi_1, \, \varphi_2}\) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.

Numerical Radius Inequalities for Certain 2 × 2 Operator Matrices

Abstract: We prove several numerical radius inequalities for certain 2 × 2 operator matrices. Among other inequalities, it is shown that if X, Y, Z, and W are bounded linear operators on a Hilbert space, then $$w\left( \left[\begin{array}{cc} X & Y \\ Z & W \end{array} \right] \right) \geq \max \left(w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right) $$ and $$w\left( \left[\begin{array}{cc}X & Y \\ Z & W\end{array} \right] \right) \leq \max \left( w(X), w(W)\right)+\frac{w(Y+Z)+w(Y-Z)}{2}. $$ As an application of a special case of the second inequality, it is shown that $$\frac{\left\Vert X\right\Vert }{2}+\frac{\left\vert \left\Vert\operatorname{Re}{X}\right\Vert -\frac{\left\Vert X\right\Vert}{2} \right\vert }{4}+\frac{ \left\vert \left\Vert \operatorname{Im}{X} \right\Vert -\frac{\left\Vert X\right\Vert}{2} \right\vert }{4} \leq w(X), $$ which is a considerable improvement of the classical inequality \({\frac{ \left\Vert X\right\Vert }{2}\leq w(X)}\) . Here w(·) and || · || are the numerical radius and the usual operator norm, respectively.

Toeplitz Operators from One Fock Space to Another

Abstract: In this paper, we study Toeplitz operators Tμ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that Tμ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if Tμ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on Tμ, we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on Cn.

Essential Self-adjointness of Magnetic Schrödinger Operators on Locally Finite Graphs

Abstract: We give sufficient conditions for essential self-adjointness of magnetic Schrodinger operators on locally finite graphs. Two of the main results of the present paper generalize recent results of Torki-Hamza.

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

Abstract: Let \({\Omega \subset{\mathbb C}^{d}}\) be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators \({T^{ u}_{g}}\) with symbol g acting on the standard weighted Bergman space \({H_ u^2}\) over Ω with weight ν. Under some conditions on the weights ν and ν0 we show that there exists C(ν, ν0) > 0, such that the Berezin transform \({\tilde{g}_{ u_{0}}}\) of g with respect to \({H^2_{ u_0}}\) satisfies: $$\label{e0}\|\tilde{g}_{ u_0}\|_\infty \leq C( u, u_0)\|T^ u_g\|,$$ for all g in a suitable class of symbols containing L∞(Ω). As a consequence we apply a result in Englis (Integr Equ Oper theory 33:426–455, 1999), to prove that the compactness of \({T^{ u}_{g}}\) is independent of the weight ν, whenever \({g \in L^{\infty}(\Omega)}\) and ν > C where C is a constant depending on (r, a, b).

Essential Normality of Homogeneous Submodules

Abstract: Let \({M \subset H(\mathbb{B})}\) be a homogeneous submodule of the n-shift Hilbert module on the unit ball in \({\mathbb{C}^{n}}\). We show that a modification of an operator inequality used by Guo and Wang in the case of principal submodules is equivalent to the existence of factorizations of the form \({[M_{z_j}^*,P_M] = (N+1)^{-1}A_j}\), where N is the number operator on \({H(\mathbb{B})}\). Thus a proof of the inequality would yield positive answers to conjectures of Arveson and Douglas concerning the essential normality of homogeneous submodules of \({H(\mathbb{B})}\). We show that in all cases in which the conjectures have been established the inequality holds and leads to a unified proof of stronger results.

On the Dunford Property (C) for Bounded Linear Operators RS and SR

Abstract: In this paper we show that if \({S\in L(X,Y)}\) and \({R\in L(Y,X),}\)X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C). We also study property (C) for R, S, RS and \({SR \in L(X)}\) in the case that R and S satisfies the operator equations RSR = R2 and SRS = S2.

Finite-Dimensional Sturm–Liouville Vessels and Their Tau Functions

Abstract: We introduce a theory of a class of finite-dimensional vessels, a concept originating from the pioneering work of Livsic (Soobshch Akad Nauk Gruzin SSSR 91(2):281–284, 1978). Our work may be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm–Liouville differentiable equations on the half axis (0,∞) with singularity at 0. We also develop a rich and interesting theory of vessels with deep connections to the notion of the τ function, arising in non linear differential equations (LDE), and to the Galois differential theory for LDEs.

Around Operator Monotone Functions

Abstract: We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if \({f(A+B)\leq f(A)+f(B)}\) for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition \({f \circ g}\) of an operator convex function f on [0,∞) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and present some applications.

On the Second Parameter of an (m, p)-Isometry

Abstract: A bounded linear operator T on a Banach space X is called an (m, p)-isometry if it satisfies the equation $${\sum_{k=0}^{m}(-1)^{k} {m \choose k}\|T^{k}x\|^{p}=0}$$ , for all $${x \in X}$$ . In this paper we study the structure which underlies the second parameter of (m, p)-isometric operators. We concentrate on determining when an (m, p)-isometry is a (μ, q)-isometry for some pair (μ, q). We also extend the definition of (m, p)-isometry, to include p = ∞ and study basic properties of these (m, ∞)-isometries.

Fractional Abstract Cauchy Problems

Abstract: This paper is concerned with fractional abstract Cauchy problems with order $${\alpha\in(1,2)}$$ . The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP 0) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP f ).

Left–Right Browder and Left–Right Fredholm Operators

Abstract: We consider left and right Browder operators, left and right Fredholm operators, spectra related with these operators, and various operator quantities.

A New Characterization for Carleson Measures and Some Applications

Abstract: We provide a new characterization for Carleson measures in terms of the L p behaviors of certain functions represented as an integration on a non-tangential cone. Applications for characterizing the boundedness and compactness of Volterra type operators from Hardy spaces to some holomorphic spaces are also presented.

Representations of Hardy Algebras: Absolute Continuity, Intertwiners and Superharmonic Operators

Abstract: Suppose $${\mathcal{T}_{+}(E)}$$ is the tensor algebra of a W*-correspondence E and H ∞(E) is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of $${\mathcal{T}_{+}(E)}$$ on a Hilbert space to ultra-weakly continuous completely contractive representations of H ∞(E) on the same Hilbert space. Our work extends the classical Sz.-Nagy–Foias functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu’s noncommutative disc algebra.

Wavelets from Laguerre Polynomials and Toeplitz-type Operators

Abstract: We study a parameterized family of Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent to a multiplication operator with a certain function. Since this function is responsible for many interesting features of these Toeplitz-type operators and their algebras, we investigate its behavior in more detail. As a by-product we obtain an interesting observation about the asymptotic behavior of true poly-analytic Bergman spaces. Isomorphisms between the Calderon-Toeplitz operator algebras and functional algebras are described and their consequences in time-frequency analysis and applications are discussed.

Compact Differences of Composition Operators in Several Variables

Abstract: When φ and ψ are linear–fractional self-maps of the unit ball BN in \({{\mathbb C}^N,N\geq 1}\), we show that the difference \({C_{\varphi}-C_{\psi}}\) cannot be non-trivially compact on either the Hardy space H2(BN) or any weighted Bergman space \({A^2_{\alpha}(B_N)}\). Our arguments emphasize geometrical properties of the inducing maps φ and ψ.

Real Linear Operator Theory and its Applications

Abstract: Real linear operators arise in a range of applications of mathematical physics. In this paper, basic properties of real linear operators are studied and their spectral theory is developed. Suitable extensions of classical operator theoretic concepts are introduced. Providing a concrete class, real linear multiplication operators are investigated and, motivated by the Beltrami equation, related problems of unitary approximation are addressed.

Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

Abstract: Let \({t\mapsto A(t)}\) for \({t\in T}\) be a CM-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here CM stands for Cω (real analytic), a quasianalytic or non-quasianalytic Denjoy–Carleman class, C∞, or a Holder continuity class C0,α. The parameter domain T is either \({\mathbb R}\) or \({\mathbb R^n}\) or an infinite dimensional convenient vector space. We prove and review results on CM-dependence on t of the eigenvalues and eigenvectors of A(t).

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

Abstract: If \({\mathfrak{A}}\) is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \({\mathbb{A}_1(1)}\), then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \({\mathbb{A}_1(1)}\), and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H∞ acting on Hardy space or on Bergman space.

Some Closed Range Integral Operators on Spaces of Analytic Functions

Abstract: Our main result is a characterization of g for which the operator \({S_g(f)(z) = \int_0^z f'(w)g(w)\, dw}\) is bounded below on the Bloch space. We point out analogous results for the Hardy space H2 and the Bergman spaces Ap for 1 ≤ p < ∞. We also show the companion operator \({T_g(f)(z) = \int_0^z f(w)g'(w) \, dw}\) is never bounded below on H2, Bloch, nor BMOA, but may be bounded below on Ap.

Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

Abstract: We study the well-posedness of the fractional differential equations with infinite delay (P2): \({D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}\), where A is a closed operator in a Banach space \({X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}\) and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P2) on Lebesgue-Bochner spaces \({L^p(\mathbb{T}, X)}\) and periodic Besov spaces \({{B} _{p,q}^s(\mathbb{T}, X)}\) .

Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Abstract: Suppose α is an orientation preserving diffeomorphism (shift) of $${{\mathbb{R}}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. We establish sufficient conditions for the Fredholmness of the singular integral operator with shift $$(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ acting on $${L^p({\mathbb{R}}_+)}$$ with 1 < p < ∞, where P ± = (I ± S)/2, S is the Cauchy singular integral operator, and $${{{W_{\alpha}f=f\circ\alpha}}}$$ is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on $${{\mathbb{R}}_+}$$ and may admit discontinuities of slowly oscillating type at 0 and ∞.

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

Abstract: The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces $${L^{p(\cdot )}(\mathbb{R} _{+},d\mu)}$$ where dμ is an invariant measure on multiplicative group $${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}$$ . (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on $${\mathbb{R}_{+}}$$ and local invertibility of singular integral operators on $${\mathbb{R}}$$ . (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.

Reducing Subspaces on the Annulus

Abstract: We study reducing subspaces for an analytic multiplication operator \({M_{z^{n}}}\) on the Bergman space \({L_{a}^{2}(A_{r})}\) of the annulus Ar, and we prove that \({M_{z^{n}}}\) has exactly 2n reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry.

Sums of Products of Toeplitz and Hankel Operators on the Dirichlet Space

Abstract: On the Dirichlet space of the unit disk, we consider operators that are finite sums of Toeplitz products, Hankel products or products of a Toeplitz operator and a Hankel operator. We characterize when such operators are equal to zero. Our results extend several known results using completely different arguments.

Scattering Theory for CMV Matrices: Uniqueness, Helson–Szegő and Strong Szegő Theorems

Abstract: We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.

Remarks on the Trotter–Kato Product Formula for Unitary Groups

Abstract: Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L2-norm, that is, one has $$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $$ for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: \({\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}\), where \({{\mathbb R}_+ := [0,\infty)}\), satisfying in addition \({{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}\) and \({{\Im{\rm m}}\,(\psi(y)) \le 0}\) for \({y \in {\mathbb R}_+}\), we prove that $$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $$ holds true uniformly on \({[0,T] i t}\) for any T > 0.

Unbounded Jacobi Matrices with a Few Gaps in the Essential Spectrum: Constructive Examples

Abstract: We give explicit examples of unbounded Jacobi operators with a few gaps in their essential spectrum. More precisely a class of Jacobi matrices whose absolutely continuous spectrum fills any finite number of bounded intervals is considered. Their point spectrum accumulates to +∞ and −∞. The asymptotics of large eigenvalues is also found.

Right Invertible Multiplication Operators and Stable Rational Matrix Solutions to an Associate Bezout Equation, I: The Least Squares Solution

Abstract: In this paper a state space formula is derived for the least squares solution X of the corona type Bezout equation G(z)X(z )= Im. Here G is a (possibly non-square) stable rational matrix function. The formula for X is given in terms of the matrices appearing in a state space representation of G and involves the stabilizing solution of an associate discrete algebraic Riccati equation. Using these matrices, a necessary and sufficient condition is given for right invertibility of the operator of multiplication by G. The formula for X is easy to use in Matlab com- putations and shows that X is a rational matrix function of which the McMillan degree is less than or equal to the McMillan degree of G. Mathematics Subject Classification (2010). Primary 47B35, 39B42; Secondary 47A68, 93B28.