Weyl-Heisenberg frames for subspaces of

A Weyl-Heisenberg frame

for allows every function to be written as an infinite linear combination of translated and modulated versions of the fixed function . In the present paper we find sufficient conditions for to be a frame for , which, in general, might just be a subspace of . Even our condition for to be a frame for is significantly weaker than the previous known conditions. The results also shed new light on the classical results concerning frames for , showing for instance that the condition A>0$">is not necessary for to be a frame for . Our work is inspired by a recent paper by Benedetto and Li, where the relationship between the zero-set of the function and frame properties of the set of functions is analyzed.

     

Frame wavelets in subspaces of

Let be a real expansive matrix. We characterize the reducing subspaces of for -dilation and the regular translation operators acting on We also characterize the Lebesgue measurable subsets of such that the function defined by inverse Fourier transform of generates through the same -dilation and the regular translation operators a normalized tight frame for a given reducing subspace. We prove that in each reducing subspace, the set of all such functions is nonempty and is also path connected in the regular -norm.

     

Invariant subspaces of the shift operator. Axiomatic approach

There is axiomatically described the class of spaces (resp. ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) z f (z) (resp. the inverse shift f(z)z^–1(f(z)–f (0))) are constructed just like the Hardy space H². It is proved that as one can take, for example, the space H¹, the disk-algebra C_A, the space U_A of all uniformly convergent power series; and as the space of integrals of Cauchy type L¹/H _– ¹ , the space VMO_A. There is also obtained an analog for the space U_A of W. Rudin's theorem on z-invariant subspaces of the space C_A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 7–26, 1981.     

Block bases of the Haar system as complemented subspaces of

It is shown that the span of , where is the Haar system in and the canonical basis of , is well isomorphic to a well complemented subspace of . As a consequence we get that there is a rearrangement of the (initial segments of the) Haar system in , any block basis of which is well isomorphic to a well complemented subspace of .

     

Invariant subspaces for polynomially hyponormal operators

We show that if is a bounded operator on a Hilbert space such that for every polynomial , then has a nontrivial invariant subspace.

     

Invariant subspaces for sequentially subdecomposable operators

In this paper, using Brown technique, we prove the Mohebi-Radjabalipour Conjecture by strengthening a slight thickness condition of the spectrum, and obtain some invariant subspace theorems. Our result contains an important known invariant subspace theorem as special cases.     

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.     

Convergence of the unitary algorithm with a unimodular Wilkinson shift

In applying the algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

     

Invariant subspaces of the quasinilpotent DT-operator

In [4] we introduced the class of DT-operators, which are modeled by certain upper triangular random matrices, and showed that if the spectrum of a DT-operator is not reduced to a single point, then it has a nontrivial, closed, hyperinvariant subspace. In this paper, we prove that also every DT-operator whose spectrum is concentrated on a single point has a nontrivial, closed, hyperinvariant subspace. In fact, each such operator has a one-parameter family of them. It follows that every DT-operator generates the von Neumann algebra of the free group on two generators.     

Blowup for

In this note we consider the global regularity of smooth solutions to the vector-valued Cauchy problem

We show that if , the gradient-blowup phenomenon occurs in finite time for suitably chosen vanishing at infinity. We also present a simple example of the -blowup solutions for for any 0$">, if .

     

Invariant subspaces of submarkovian semigroups

We characterize the invariance under a submarkovian semigroup of a measurable subset by capacity conditions on its boundary.      

Invariant subspaces of Toeplitz operators

This paper is devoted to the problem of the existence of invariant subspaces for Toeplitz operators. Let be a Lipschitzian arc in the plane and let f be a non-constant continuous functions on the unit circumference. It is proved that if there exists an open circle such that and if the modulus of continuity _f of the function f satisfies the condition then the Toeplitz operator T_f in the Hardy space H² has a nontrivial hyperinvariant subspace. For the proof of this theorem one makes use of the Lyubich-Matsaev theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 170–179, 1983.I express my deep gratitude to E. M. Dyn'kin for useful discussions.