We characterize when a Hankel operator and a Toeplitz operator have a compact commutator. Let dσ(w) be the normalized Lebesgue measure on the unit circle ∂D. The Hardy space H is the subspace of L(∂D, dσ), denoted by L, which is spanned by the space of analytic polynomials. So there is an orthogonal projection P from L onto the Hardy space H, the so-called Hardy projection. Let f be in L∞. The ...

Except for the Toeplitz and Hankel matrices, the common patterned matrices for which the limiting spectral distribution (LSD) are known to exist share a common property–the number of times each random variable appears in the matrix is (more or less) the same across the variables. Thus it seems natural to ask what happens to the spectrum of the Toeplitz and Hankel matrices when each entry is sca...

Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computations in Sciences, Engineering and Signal and Image Processing. The four matrix classes have distinct features, but in [P90] we showed that Vandermonde and Hankel multipliers transform all these structures into each other and proposed to employ this property in order to extend any suc...

Gaussian elimination is a standard tool for computing triangular factorizations for general matrices, and thereby solving associated linear systems of equations. As is well-known, when this classical method is implemented in finite-precision-arithmetic, it often fails to compute the solution accurately because of the accumulation of small roundoffs accompanying each elementary floating point op...

The papers [MRT05], [CGS07], [XXG12], and [XXCB14] combine the techniques of the Fast Multipole Method of [GR87], [CGR98] with the transformations of matrix structures, traced back to [P90]. The resulting numerically stable algorithms approximate the solutions of Toeplitz, Hankel, Toeplitz-like, and Hankel-like linear systems of equations in nearly linear arithmetic time, versus the classical c...

We apply and extend some well known and some recent tech niques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is computa tions with structured matrices and solving a system of polynomial equa tions In the rst part of our paper we extend the Toeplitz and Hankel structures of matrices and some of their known properties ...

Abstract Suppose that Tn is a Toeplitz matrix whose entries come from a sequence of independent but not necessarily identically distributed random variables with mean zero. Under some additional tail conditions, we show that the spectral norm of Tn is of the order √ n log n. The same result holds for random Hankel matrices as well as other variants of random Toeplitz matrices which have been st...

In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on ∂D. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz oper...

In this article, we characterize the sufficient and necessary conditions for positiveness of operator matrices with Toeplitz little Hankel operators on Bergman space. Further, explore some to be normal unitary.

We consider the problem of solving least squares problems involving a matrix M of small displacement rank with respect to two matrices Z 1 and Z 2. We develop formulas for the generators of the matrix M (H) M in terms of the generators of M and show that the Cholesky factorization of the matrix M (H) M can be computed quickly if Z 1 is close to unitary and Z 2 is triangular and nilpotent. These...