Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.     

Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators

The Keldysh theorem is generalized to an arbitrary closed operator that is not necessarily close to self-adjoint operators and has a resolvent of Schatten–von Neumann class S _p . Based on this theorem, conditions of spectrum localization are obtained for certain classes of non-self-adjoint differential operators.     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

On weak supercyclicity II

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbertspace operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l-sequentially supercyclic, and (iii) weak l-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators: (iv) the point spectrum of the normed-space adjoint of a power bounded supercyclic operator is either empty or is a singleton in the open unit disk, (v) weak l-sequential supercyclicity coincides with supercyclicity for compact operators, and (vi) every compact weakly l-sequentially supercyclic operator is quasinilpotent.     

Models for n-tuples of noncommuting operators

This paper proves versions of the Rota model theorem, the de Branges-Rovnyak model theorem, and the coisometric extension theorem for n-tuples of not necessarily commuting operators. This generalizes the work of A. E. Frazho (J. Funct. Anal.48 (1982), 1–11) for pairs of operators. The methods involve applying the single operator results to matrices of operators.     

Orthogonal and Skew-Symmetric Operators in Real Hilbert Space

In the theory of traces on operator ideals, it is desirable to treat not only the complex case. Several proofs become much easier when the underlying operators are represented by real matrices. Motivated by this observation, we prove two theorems which, to the best of our knowledge, are not available in the real setting: (1) every operator is a finite linear combination of orthogonal operators, and (2) every skew-symmetric compact operator S is a commutator [A, T], where certain properties of S are inherited to T. In our opinion, theses results are interesting for their own sake. They will also be used in future studies of trace theory by the second-named author.     

Measure-compact operators, almost compact operators, and linear functional equations in L p

Under study are the measure-compact operators and almost compact operators in L _p . We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in L _p , we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in L _p with almost compact or measure-compact operators to equivalent linear integral equations in L _p with quasidegenerate Carleman kernels.     

一类不可分解Banach空间上线性算子的谱性质

给出一类不可分解的∑_e~1型Banach空间上线性算子(不一定有界)的谱结构,并讨论这种空间上生成C_0群或C_0半群的线性算子的有界性、特殊的谱性质和谱结构,还给出这种空间上闭算子是有界算子的一个充分条件。     

Some problems on narrow operators on function spaces

It is known that if a rearrangement invariant (r.i.) space E on [0, 1] has an unconditional basis then every linear bounded operator on E is a sum of two narrow operators. On the other hand, for the classical space E = L ₁[0, 1] having no unconditional basis the sum of two narrow operators is a narrow operator. We show that a Köthe space on [0, 1] having “lots” of nonnarrow operators that are sum of two narrow operators need not have an unconditional basis. However, we do not know if such an r.i. space exists. Another result establishes sufficient conditions on an r.i. space E under which the orthogonal projection onto the closed linear span of the Rademacher system is a hereditarily narrow operator. This, in particular, answers a question of the first named author and Randrianantoanina (Problem 11.9 in [Popov M., Randrianantoanina B., Narrow Operators on Function Spaces and Vector Lattices, de Gruyter Stud. Math., 45, Walter de Gruyter, Berlin, 2013]).     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

A Generalization of Tyuriemskih's Lethargy Theorem and Some Applications

Tyuriemskih's Lethargy Theorem is generalized to provide a useful tool for establishing when a sequence of (not necessarily) linear operators that converges point wise to the identity operator actually converges arbitrarily slowly. Then this generalization is used to answer affirmatively a 2010 conjecture of ours as well as establishing that all of the classical operators of Bernstein, Hermite-Fejer, Landau, Fejer, and Jackson converge arbitrarily slowly to the identity operator (and not just almost arbitrarily slowly as we established in 2010).     

Approximation by chaotic operators and by conjugate classes

We show that every bounded linear operator on a separable, infinite-dimensional Hilbert space H is the sum of two operators in the norm-closure of the set of operators on H that are chaotic in the sense of Devaney. We also observe that the density of several classes of cyclic operators, with respect to the strong operator topology, may be derived from a result by Hadwin et al. (Proc Amer. Math. Soc. 76 (1979) 250-252).     

Form sums of nonnegative selfadjoint operators

Summary The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.</o:p>     

On the duality theorem of bounded S-decomposable operators

A class of linear operators, more general than that of the decomposable operators, here referred to as S-decomposable operators, was introduced in [10]. It was shown in [11] that the dual of an S-decomposable operator was again S-decomposable. Our main result, in this paper, is the converse implication, namely, if the dual $\text{T1}$ of a bounded linear operator T is S-decomposable then T is S-decomposable.     

Reflexive linear transformations

A linear operator A is called reflexive if the only operators that leave invariant the invariant subspaces of A are the operators in the weak closure of the algebra of polynomials in A. In this note we completely characterize reflexive operators on finite-dimensional spaces.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

Weyl-Type Theorems on Banach Spaces Under Compact Perturbations

In this paper, we study Browder-type and Weyl-type theorems for operators \(T+K\) defined on a Banach space X, where K is (a non necessarily commuting) compact operator on X. In the last part, the theory is exemplified in the case of isometries, analytic Toeplitz operators, semi-shift operators, and weighted right shifts.