On the hyperinvariant subspace problem III

In two recent papers (Foias and Pearcy, J. Funct. Anal., in press, Hamid et al., Indiana Univ. Math. J., to appear), the authors reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every C₀₀-(BCP)-operator that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). In this note, we continue this study by showing, with the help of a new equivalence relation, that every operator whose spectrum is uncountable, as well as every nonalgebraic operator with finite spectrum, has a hyperlattice (i.e., lattice of hyperinvariant subspaces) that is isomorphic to the hyperlattice of a C₀₀, quasidiagonal, (BCP)-operator whose spectrum is the closed unit disc.     

Quasidiagonality and the hyperinvariant subspace problem

In a sequence of recent papers, [11], [13], [9] and [5], the authors (together with H. Bercovici and C. Foias) reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every C ₀₀-(BCP)-contraction that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). An essential ingredient in this reduction was the introduction of two new equivalence relations, ampliation quasisimilarity and hyperquasisimilarity, defined below. This note discusses the question whether, by use of these relations, a further reduction of the hyperinvariant subspace problem to the much-studied class (N + K) (defined below) might be possible.     

On the filling in holes problem for operator matrices

We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces.     

Algebraic isomorphisms and strongly double triangle subspace lattices

Let D={{0},K,L,M,X} be a strongly double triangle subspace lattice on a non-zero complex reflexive Banach space X, which means that at least one of three sums K + L, L + M and M + K is closed. It is proved that a non-zero element S of AlgD is single in the sense that for any A,B∈AlgD, either AS = 0 or SB = 0 whenever ASB = 0, if and only if S is of rank two. We also show that every algebraic isomorphism between two strongly double triangle subspace lattice algebras is quasi-spatial.     

On the eigenvalue problem for a particular class of finite Jacobi matrices

A function f with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of f, first of all the Bessel functions of first kind. A compact formula in terms of the function f is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function f in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors.     

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

Inversion of regular analytic matrix functions: Local Smith form and subspace duality

This paper analyzes the relation between the local rank-structure of a regular analytic matrix function and the one of its inverse function. The local rank factorization (lrf) of a matrix function is introduced, which characterizes extended canonical systems of root functions and the local Smith form. An interpretation of the local rank factorization in terms of Jordan chains and Jordan pairs is provided. Duality results are shown to hold between the subspaces associated with the lrf of the matrix function and the one of its reduced adjoint.     

On closures of joint similarity orbits

For an n-tuple T=(T₁,..., T_n) of operators on a Hilbert spacexxHx, the joint similarity orbit of T isxxSx(T)={VTV^–1 =(VT₁V^–1,...,VT_nV^–1): V is invertible onxxHx}. We study the structure of the norm closure ofxxSx, both in the case when T is commutative and when it is not. We first develop a Rota-model for the Taylor spectrum and use it to study n-tuples with totally disconnected Taylor spectrum, in particular quasinilpotent ones. We then consider limits of nilpotent n-tuples, and of normal n-tuples. For noncommuting n-tuples, we present a number of surprising facts relating the closure ofxxSx(T) to the Harte spectrum of T and the lack of commutativity of T. We show that a continuous function which is constant onxxSx(T) for all T must be constant. We conclude the paper with a detailed study of closed similarity orbits.Research partially supported by grants from the National Science Foundation.     

Generalized hermitian operators

In this paper, the concept of generalized hermitian operators defined on a complex Hilbert space is introduced. It is shown that the spectrums and the Fredholm fields of generalized hermitian operators are both symmetric with respect to the real axis. Some other results on generalized hermitian operators are obtained.     

On the eigenvalues of some tridiagonal matrices

A solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by William Trench. The method presented can be generalizable to other problems.     

The spectral picture and the closure of the similarity orbit of strongly irreducible operators

An operator on a complex, separable, infinite dimensional Hilbert space is strongly irreducible if it does not commute with any nontrivial idempotent. This article answers the following questions of D. A. Herrero: (i) Given an operatorT with connected spectrum, can we find a strongly irreducible operatorL such that they have same spectral picture? (ii) When we use a sequence of irreducible operators to approximateT, can the approximation be the “most economic”? i.e., does there exist a strongly irreducible operatorL such thatT ∈S(L) ^? (the closure of the similarity orbit ofL)? It is shown that the answer for the two questions is yes.     

On a subspace metric based on matrix rank

The metric between subspaces M,N⊆C_n,1, defined by δ(M,N)=rk(P_M-P_N), where rk(·) denotes rank of a matrix argument and P_M and P_N are the orthogonal projectors onto the subspaces M and N, respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ(M,N) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.     

Filling in the holes of the semi-Fredholm domain

We study the following problem: given a set of holes in the semi-Fredholm domain of an operator, is there an invariant subspace of the operator such that the spectrum of the restriction is equal to the spectrum of the operator together with the set of holes?     

Square-root Householder subspace tracking

A class of singular value decomposition (SVD)-type subspace trackers based on the overdetermined row-Householder principle is introduced. These algorithms are maximally fast with a dominant operations count of 3Nr multiplications per time update. They can be regarded as square-root forms of previously introduced conventional fast subspace trackers and offer interesting features such as perfectly orthonormal basis estimates, lowest dynamic range requirements, and highest numerical robustness and stability. Several variants of the method are proposed and studied experimentally.     

On the inverse mean first passage matrix problem and the inverse M-matrix problem

The inverse mean first passage time problem is given a positive matrix M∈R^n,n, then when does there exist an n-state discrete-time homogeneous ergodic Markov chain C, whose mean first passage matrix is M? The inverse M-matrix problem is given a nonnegative matrix A, then when is A an inverse of an M-matrix. The main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other. In so doing we extend earlier results of Tetali and Fiedler.     

The effect of non-optimal bases on the convergence of Krylov subspace methods

Summary There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of non-optimal methods have been shown to converge in many situations, often with small delay, but not in others. We explore the question of what is the effect of having a non-optimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and non-optimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. Our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results.This revised version was published online in June 2005 due to a typesetting mistake in the footnote on page 7.     

Multilinear Forms of Hilbert Type and Some Other Distinguished Forms

We give some new examples of bounded multilinear forms on the Hilbert spaces ℓ₂ and L₂ (0, ∞). We characterize those which are compact or Hilbert-Schmidt. In particular, we study m-linear forms (m ≥ 3) on ℓ₂ which can be regarded as the multilinear analogue of the famous Hilbert matrix. We also determine the norm of the permanent on where      

Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.