Non-compact Operators Research Articles

A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces

This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection $$\mathcal{C}(X)$$ (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from $$\mathcal{C}(X)$$ onto $$\mathcal{F}(\Omega)$$ is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and $$\mathcal{F}(\Omega)$$ the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both $$E_\mathcal{C}=\overline{J\mathcal{C}-J\mathcal{C}}$$ and $$E_\mathcal{K}=\overline{J\mathcal{K}-J\mathcal{K}}$$ are Banach lattices and $$E_\mathcal{K}$$ is a lattice ideal of $$E_\mathcal{C}$$ . The quotient space $$E_\mathcal{C}/E_\mathcal{K}$$ is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and $$(FQJ)\mathcal{C}$$ which is a closed cone is contained in the positive cone of C(K), where $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C}/E_\mathcal{K}}$$ is the quotient mapping and $$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given: Let BX be the closed unit ball of a Banach space X, then $$\mu(T)=\mu(T(B_X))=\parallel(FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$

Regularization of linear ill-posed problems involving multiplication operators

ABSTRACT We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-posed equations with non-compact operators less is known if the data are blurred by white noise. We develop a theory for spectral regularization with emphasis on this case. In this context, we highlight several aspects, in particular, we discuss the intrinsic degree of ill-posedness in terms of rearrangements of the multiplier function. Moreover, we address the required modifications of classical regularization schemes in order to be used for non-compact statistical problems, and we also introduce the concept of the effective ill-posedness of the operator equation under white noise. This study is concluded with prototypical examples for such equations, as these are deconvolution equations and certain final value problems in evolution equations.

Krein–Rutman type property and exponential separation of a noncompact operator

We investigate exponentially separated property for a noncompact linear operator T T on a Banach space. We obtain the relationship between exponentially separated property and the well-known Krein–Rutman type property for a noncompact operator. Under the assumption of an essential spectral gap, we prove that any u u -bounded operator T T with a reproducing cone admits the exponentially separated property and, hence, is of Krein–Rutman type automatically. We also establish an amenable sufficient condition for the exponentially separated property of some degenerate linear parabolic systems.

Approximate controllability of a fractional stochastic partial integro-differential systems via noncompact operators

Objective: in this article, we discuss the approximate controllability problems of a new class of fractional impulsive stochastic partial integro-differential systems in separable Hilbert spaces. Methods: by applying the fractional calculus, the measure of noncompactness, properties of fractional resolvent operators and fixed point theorems. Results: we prove our main results without the hypotheses of compactness on the operator generated by the linear part of systems. Instead we suppose that the nonlinear term only satisfies a weakly compactness condition. Conclusion: the approximate controllability for the control systems with noncompact operators is established. Finally, an example is given for the illustration of the obtained theoretical results.

Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces

ABSTRACTWe studied some topological properties of the space of bounded Volterra-type integral operators on generalized Fock spaces with weight function growing faster than the classical Gaussian function. In particular, the connected and path-connected components, isolated and essentially isolated points of the space are described. The result shows that while the space contains all the non-compact operators as its isolated points, it fails to admit essentially isolated points.

Global Attractor for a Class of Sixth-Order Viscous Cahn-Hilliard Equation in an Unbounded Domain

In this paper, we consider the existence of global attractor for a class of sixth-order Cahn-Hilliard equation with a nonlinear diffusion and viscous effects in an infinite domain. Due to the noncompactness of operators, we use weighted Sobolev spaces to prove that the semigroup generated by the equation has the global attractor in a suitable space.

Acoustic vibration problem for dissipative fluids

In this paper we analyze a finite element method for solving a quadratic eigenvalue problem derived from the acoustic vibration problem for a heterogeneous dissipative fluid. The problem is shown to be equivalent to the spectral problem for a noncompact operator and athorough spectral characterization is given. The numerical discretization of the problem is based on Raviart-Thomas finite elements. The method is proved to be free of spurious modes and to converge with optimal order. Finally, we report numerical tests which allow us to assess the performance of the method.

Interpolation and extrapolation of strictly singular operators between Lp spaces

We study the interpolation and extrapolation properties of strictly singular operators between different Lp spaces. To this end, the structure of strictly singular non-compact operators between Lp−Lq spaces is analyzed. Among other things, we clarify the relation between strict singularity and the L-characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators.

Singular Values of Weighted Composition Operators and Second Quantization

We study a semigroup of weighted composition operators on the Hardy space of the disk $H^2(\mathbb{D})$, and more generally on the Hardy space $H^2(U)$ attached to a simply connected domain $U$ with smooth boundary. Motivated by conformal field theory, we establish bounds on the singular values (approximation numbers) of these weighted composition operators. As a byproduct we obtain estimates on the singular values of the restriction operator (embedding operator) $H^2(V) \to H^2(U)$ when $U \subset V$ and the boundary of $U$ touches that of $V$. Moreover, using the connection between the weighted composition operators and restriction operators, we show that these operators exhibit an analog of the Fisher-Micchelli phenomenon for non-compact operators.

Cordial Volterra Integral Equations with Vanishing Delays*

Cordial Volterra integral equations (CVIEs) from some applications models associated with a noncompact cordial Volterra integral operator are discussed in the recent years. A lot of real problems are effected by a delayed history information. In this paper we investigate some properties of cordial Volterra integral operators influenced by a vanishing delay. It is shown that to replicate all eigenfunctions , or , the vanishing delay must be a proportional delay. For such a linear delay, the spectrum, eigenvalues and eigenfunctions of the operators and the existence, uniqueness and solution spaces of solutions are presented. For a nonlinear vanishing delay, we show a necessary and sufficient condition such that the operator is compact, which also yields the existence and uniqueness of solutions to CVIEs with the vanishing delay.

The Markov–Stieltjes transform on Hardy and Lebesgue spaces

ABSTRACTWe prove that the Markov–Stieltjes transform is a bounded non compact Hankel operator on Hardy space with Hilbert matrix with respect to the standard Schauder basis of and a bounded non-compact operator on Lebesgue space for and obtain estimates for its norm in this spaces. It is shown that the Markov–Stieltjes transform on is unitary equivalent to the Markov–Stieltjes transform on . Inverse formulas and operational properties for this transform are obtained.

On the functional Hodrick–Prescott filter with non-compact operators

Abstract We study a version of the functional Hodrick–Prescott filter in the case when the associated operator is not necessarily compact but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case when the underlying distribution of the data is Gaussian.

Bifurcation problems for noncompact operators

Bifurcation problems for noncompact operators

On Solvability of one Class of Nonlinear Integral-differential Equation with Hammerstein Non-compact Operator Arising in a Theory of Income Distribution

On Solvability of one Class of Nonlinear Integral-differential Equation with Hammerstein Non-compact Operator Arising in a Theory of Income Distribution

On the solvability of one class of nonlinear integral equations in L 1(0,+∞)

We study the problem of constructing a solution that is positive, integrable, and essentially bounded on (0,+∞) to one class of nonlinear Hammerstein integral equations with noncompact operator in the critical case.

On solvability of a class of nonlinear second order integro-differential equations in the space W 1 2 (ℝ+)

In this paper we study an initial-boundary value problem for a second order nonlinear integro-differential equation with Hammerstein type noncompact operator on the positive semi-axis. This problem has a direct application in the nonlocal problems of physical kinetics. We prove the existence of a nonnegative (nontrivial) solution of this equation in the space W12 (ℝ+). Some special examples of these equations are also discussed, which represent independent theoretical and applied interest.

Essential Norm of Toeplitz Operators on the Fock Spaces

In this paper, we show that, on the generalized Fock space \({F^p(\varphi)}\) with \({1 < p < \infty}\) , the essential norm of a noncompact Toeplitz operator \({T_\mu}\) with \({|\mu|}\) being a Fock–Carleson measure equals its distance to the set of compact Toeplitz operators. Moreover, the distance is realized by infinitely many compact Toeplitz operators. Our approach is also available on the Bergman space setting.

On the Solvability of One Class of Nonlinear Integral Equations with a Noncompact Hammerstein–Stieltjes-Type Operator on the Semiaxis

We study a class of nonlinear integral equations with a noncompact operator of the Hammerstein–Stieltjes-type on the semiaxis. The existence of positive solutions is proved in various function spaces by using the factorization methods and specially chosen successive approximations.

Nonexistence of certain universal polynomials between Banach spaces

A well-known result due to W.B. Johnson (1971) asserts that the formal identity operator from ℓ1 into ℓ∞ is universal for the class of non-compact operators between Banach spaces. We show that there is neither a universal non-compact polynomial nor a universal non-unconditionally converging polynomial between Banach spaces.

Approximation numbers of composition operators on the [formula omitted] space of Dirichlet series

By a theorem of Gordon and Hedenmalm, φ generates a bounded composition operator on the Hilbert space H2 of Dirichlet series ∑nbnn−s with square-summable coefficients bn if and only if φ(s)=c0s+ψ(s), where c0 is a nonnegative integer and ψ a Dirichlet series with the following mapping properties: ψ maps the right half-plane into the half-plane Res>1/2 if c0=0 and is either identically zero or maps the right half-plane into itself if c0 is positive. It is shown that the nth approximation numbers of bounded composition operators on H2 are bounded below by a constant times rn for some 0<r<1 when c0=0 and bounded below by a constant times n−A for some A>0 when c0 is positive. Both results are best possible. The case when c0=0, ψ is bounded and smooth up to the boundary of the right half-plane, and sup⁡Reψ=1/2, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes Sp. For φ(s)=c1+∑j=1dcqjqj−s with qj independent integers, it is shown that the nth approximation number behaves as n−(d−1)/2, possibly up to a factor (log⁡n)(d−1)/2. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for H2 using estimates of solutions of the ∂¯ equation. Finally, by a transference principle from H2 of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.