Finite Rank Theorem Research Articles

D. Luecking’s Finite Rank Theorem for Toeplitz Operator, Benedicks’ Theorem on the Heisenberg Group, and Uncertainty Principle for the Fourier–Wigner Transform

D. Luecking’s Finite Rank Theorem for Toeplitz Operator, Benedicks’ Theorem on the Heisenberg Group, and Uncertainty Principle for the Fourier–Wigner Transform

Connection Matrices and the Definability of Graph Parameters

In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers.

Some weighted estimates for the ∂̅-equation and a finite rank theorem for Toeplitz operators in the Fock space

We consider the ∂ ¯ -equation in C 1 in classes of functions with Gaussian decay at infinity. We prove that if the right-hand side of the equation is majorated by exp ( − q | z | 2 ) , with some positive q, together with derivatives up to some order, and is orthogonal, as a distribution, to all analytical polynomials, then there exists a solution with decays, together with derivatives, as exp ( − q ′ | z | 2 ) , for any q ′ < q / e . This result carries over to the ∂ ¯ -equation in classes of distributions, again, with Gaussian decay at infinity, in some precisely defined sense. The properties of the solution are used further on to prove the finite rank theorem for Toeplitz operators with distributional symbols in the Fock space: the symbol of such operator must be a combination of finitely many δ-distributions and their derivatives. The latter result generalizes the recent theorem on finite rank Toeplitz operators with symbols–functions.

The Finite Rank Theorem for Toeplitz Operators on the Fock Space

We consider Toeplitz operators on the Fock space, under rather general conditions imposed on the symbols. We prove that if a Toeplitz operator has finite rank, then the operator and its symbol are zero. Our argument is somewhat different from the ones used previously for finite rank theorem, and it enables us to get rid of the compact support condition and even allows for a certain growth of the symbol.

A RANK THEOREM FOR NONLINEAR SEMI-FREDHOLM OPERATORS BETWEEN TWO BANACH MANIFOLDS

In this article the concept of local conjugation of a C 1 mapping between two Banach manifolds is introduced. Then a rank theorem for nonlinear semi-Fredholm operators between two Banach manifolds and a finite rank theorem are established in global analysis.

Rank theorems of operators between Banach spaces

LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ∥ T0 # (T - T0 ∥ < 1, being a generalized inverse ofT, is presented, and hence, a rank theorem of operators onE intoF is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.

(1.2) INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

Let E and F be Banach spaces and f non-linear C1 map from E into F. The main result is Theorem 2.2, in which a connection between local conjugacy problem of f at x0∈ E and a local fine property of f′(x) at x0 (see the Definition 1.1 in this paper) are obtained. This theorem includes as special cases the two known theorems: the finite rank theorem and Berger's Theorem for non-linear Fredholm operators. Moreover, the theorem gives rise the further results for some non-linear semi-Fredholm maps and for all non-linear semi-Fredholm maps when E and F are Hilbert spaces. Thus Theorem 2.2 not only just unifies the above known theorems but also really generalizes them.

On the analogue of Corner's finite rank theorem for modules over valuation domains

On the analogue of Corner's finite rank theorem for modules over valuation domains