Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let denote the class of quasinilpotent quasiaffinities Q in such that Q ^* Q has an infinite dimensional reducing subspace M with Q ^* Q| _M compact. It was known that if every quasinilpotent operator in has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in . The purpose of this paper is to provide sufficient conditions for elements in to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.      

Some Algebraic Operators and the Invariant Subspace Problem

We consider the resolvent algebra ${R_A=\{T\in\mathcal{L} (X):\sup_{m \geq 0}\|(1+\,mA)T(1+\,mA)^{-1}\| < \infty \}}$ , and Deddens?? algebra ${B_A= \{T\in \mathcal{B}(H) : \sup_{n\geq 0}\|A^nTA^{-n}\|<\infty\}}$ . It is shown that both R _A and B _A?CI possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R _A whenever R _A ?? {A}??. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R _A when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R _A contains properly {A}?? whenever A has an eigenvalue other than zero.     

Invariant and Hyperinvariant Subspace Lattices of Operators $$J^alpha otimes B$$ in Sobolev Spaces

Mathematical Notes - Let .     

Quasinilpotent Operators and the Invariant Subspace Problem

In this paper we construct quasinilpotent operators withoutnontrivial invariant subspaces. As well as being interestingin its own right, we believe that this is a sensible (perhapsnecessary) first step for addressing the difficult problem ofconstructing topologically simple Banach algebras.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Approximation Properties and Some Classes of Operators

The following questions and close problems are studied.(i) Is it true that T is p-nuclear provided that T ^** is p-nuclear? (ii) Is it true that Tis dually p-nuclear provided that T ^* is p-nuclear? (iii) Is it true that if T ^*is compactly factorable in the space l _p, then T is (strictly) factorable in the space l _p'? Here, T ^* is the adjoint operator of a bounded operator T:X Yin Banach spaces X and Y. Bibliography: 30 titles.     

Hyperinvariant Closed Ideals for a Finitely Quasinilpotent Collection of Operators

Mathematical Notes - In this paper, it is proved that if $$\mathscr C\ne\{0\}$$ is a collection of continuous operators with modulus on an $$\ell_p$$ -space ( $$1\le p&lt;\infty$$ ) that is...     

Characterizations of Some Matrix Classes of Quasi-Homogeneous Operators

In this paper, we give a general characterization of a class of matrices of quasi-homogeneous operators between topological vector spaces.     

几类增算子的不动点定理及其应用

讨论了C[I,E]和L_p[I,E]中几类增算子,得到了若干新的不动点定理,并将其应用于证明非线性微分方程唯一解的存在性.     

Characteristic Classes for Difference Operators

We introduce and describe the characteristic class of a difference operator over the difference field (k((t)),). Here k is an algebraically closed field of characteristic zero and is the k-linear automorphism of k((t)) defined by (t)=t/(1+t). The approach is based on the characterization of simple difference operators in terms of their eigenvalues.     

Extension Problem and Harnack's Inequality for Some Fractional Operators

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy–Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H ^σ = (? Δ + |x|²)^σ to deduce a Harnack's inequality. A pointwise formula for H ^σ f(x) and some maximum and comparison principles are derived.     

Some Sharp Norm Estimates in the Subspace Perturbation Problem

We discuss the spectral subspace perturbation problem for a self-adjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tan Θ Theorem. We also extend the Davis–Kahan tan 2Θ Theorem to the case of some unbounded perturbations.     

On Invariance of Some Classes of Holomorphic Functions Under Integrodifferential Operators

The following classes of functions analytic in the unit disk are considered:

几类非线性二元算子方程的迭代解法及其应用

利用锥与半序理论和单调迭代技巧，讨论几类非线性二元算子方程解的存在唯一性，并给出迭代序列收敛速度的估计，改进和推广了某些已有结果．最后给出所得结果的应用。     

Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99 Y.A. Abramovich: 1945–2003 The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846     

Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.     

Resolvent Estimates for Operators Belonging to Exponential Classes

For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s _n (A) = O(exp(-an^α)), where s _n (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)⁻¹ of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).      

A Class of C ·0-Contractions: Hyperinvariant Subspaces and Intertwining Operators

We consider a class of C_·0-contractions that is a generalization of the class of C_·0-contractions with finite defect indices. Some results of Uchigama and Wu for C_·0-contractions with finite defect indices are generalized: the lattices of hyperinvariant subspaces of such a contraction T is isomorphic to that of its Jordan model and is generated by subspaces of the form Ker ϕ(T) and Ran ϕ(T), where ϕ ∈ H^∞. The form of the inverse to an isomorphism of the invariant subspace lattices given by an intertwining quasiaffinity is also studied. Next, for C_·0-contractions in question, the characteristic disc related to the lattice of invariant subspaces is computed. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 48–62.     

On Certain Positivity Classes of Operators

A real square matrix A is called a P-matrix if all its principal minors are positive. Such a matrix can be characterized by the sign non-reversal property. Taking a cue from this, the notion of a P-operator is extended to infinite dimensional spaces as the first objective. Relationships between invertibility of some subsets of intervals of operators and certain P-operators are then established. These generalize the corresponding results in the matrix case. The inheritance of the property of a P-operator by the Schur complement and the principal pivot transform is also proved. If A is an invertible M-matrix, then there is a positive vector whose image under A is also positive. As the second goal, this and another result on intervals of M-matrices are generalized to operators over Banach spaces. Towards the third objective, the concept of a Q-operator is proposed, generalizing the well known Q-matrix property. An important result, which establishes connections between Q-operators and invertible M-operators, is proved for Hilbert space operators.