Trace formulas for a class of Toeplitz-like operators

LetP _T denote projection onto the space of entire functions of exponential type ≦T which are square summable on the line relative to a measuredΔ and letG denote multiplication by a suitably restricted complex valued function,g. For a reasonably large class of measuresdΔ, which includes Lebesgue measuredγ, it is shown that trace {(P _TGP_T)ⁿ−P_TGⁿP_T} tends boundedly to a limit asT↑∞ and that the limit isindependent of the choice ofdΔ within the permitted class. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special casedΔ=dγ using a different formalism.     

Trace formulas for blocks of Toeplitz-like operators

In this paper we study trace formulas for a class of operators of the form Π_T-GΠ_T in which G designates multiplication by a suitable restricted d × d matrix valued function G(γ) of γ?R¹and Π_T stands for the diagonal d × d matrix (δ_ijP_T) of orthogonal projections P_T of L²(R¹, dδ) onto the space I^T(dδ) of entire functions of exponential type ?T which are square summable on the line relative to the measure $\text{dδ(γ) = ¦h(γ)¦}{}^2\text{dγ}$. It is shown that, for a reasonably large class of h,

$\text{lim}\text{T↑∞}\text{trace}\text{[(?}{}_{\mathrm{T}}\text{G?}{}_{\mathrm{T}}\text{)}{}^{\mathrm{n}}\text{? ?}{}_{\mathrm{T}}\text{G}{}^{\mathrm{n}}\text{?}{}_{\mathrm{T}}\text{]}$

exists and is independent of the choice of h within the permitted class. These results are then used to study the asymptotic behavior, as T ↑ ∞, of the determinant of I ? Π_TGΠ_T.     

Trace formulas for a class of subnormal tuples of operators

Trace formulas are established for the product of commutators related to subnormal tuple of operators (S ₁,...,S _n ) with minimal normal extension (N ₁,...,N _n ) satisfying conditions that sp(S _j )/sp(N _j ) is simply-connected with smooth boundary Jordan curve sp(N _i ) and [S _j ^* ,S _j ]^1/2 L ¹,j=1, 2,...,n.Some complete unitary invariants related to the trace formulas are found.This work is supported in part by NSF Grant no. DMS-9101268.     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

Partial indices for Toeplitz-like operators

The concept of partial indices, which is usually associated with matrix-functions, is defined for operators A satisfying rank (AU-VA)< for some fixed U and V (Toeplitz-like operators). A survey of the properties of partial indices is presented, some examples, and the connection with Kronecker indices as well. An application concerning Toeplitz plus Hankel matrices is supplemented.     

Trace formulas for non-self-adjoint operators with discrete spectrum

We prove a trace formula, which is an abstract analog of the well-known Gel'fand-Levitan formula.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 228–236, 1988.     

Trace formulas for relative Schatten class perturbations

We derive trace formulas for a pair of self-adjoint operators $H+V$ and H under the assumption that ${(H?\mathrm{i})}^{?1}V$ is in a Schatten class. This extends the trace formulas of [8], where V alone is assumed to be in a Schatten class. Our trace formulas apply, in particular, in the setting of differential operators and are based on Taylor-like approximations of operator functions. This significantly improves non-Taylor based trace formulas of [10].     

Trace formulas for almost commuting operators,cyclic cohomology and subnormal operators

Some formulas related to cyclic cohomology for the trace of the product of commutators are established. A simple complete unitary invariant for some subnormal operator with simply connected spectrum is found.This work is supported in part by NSF grant.     

Trace class norm inequalities for localization operators

We give sharp estimates on the norms in the trace class of localization operators in terms of their symbols.     

Trace formulas for nuclear operators in spaces of Bochner integrable functions

The paper is devoted to trace formulas for nuclear operators in spaces of Bochner integrable functions. We characterise nuclearity for integral operators on such spaces and develop a trace formula for general kernels applying vector-valued maximal functions.     

Trace formulas for non-self-adjoint periodic Schrödinger operators and some applications

Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in L²(R) associated with Dirichlet eigenvalues was proved in [Differential Integral Equations 14 (2001) 671-700]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas, we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential Ke2ix, where K∈C.     

Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

We present simple trace formulas for Hecke operators Tk(p) for all p>3 on Sk(Γ₀(3)) and Sk(Γ₀(9)), the spaces of cusp forms of weight k and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over Fq, when . As an application, we use these formulas to provide a simple expression for the Fourier coefficients of η⁸(3z), the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.     

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H=AS⁺+A^-S^-+B (with S^± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E₊], E_-<E₊, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by , 0?E_-<E₊.Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.