In this paper, we study the relationship between the acquisition of time-harmonic seismic data and the Dirichlet-to-Neumann map for the Helmholtz equation in dimension n ≥ 3. This relationship is established through the introduction of a single-layer potential operator. We analyze its properties with a view to so-called iterative full waveform inversion based on the Hilbert-Schmidt norm, that i...

Let A be a m × m complex matrix with zero trace. Then there are m ×m matrices B and C such that A = [B,C] and ‖B‖‖C‖2 ≤ (logm + O(1))‖A‖2 where ‖D‖ is the norm of D as an operator on `2 and ‖D‖2 is the Hilbert–Schmidt norm of D. Moreover, the matrix B can be taken to be normal. Conversely there is a zero trace m × m matrix A such that whenever A = [B,C], ‖B‖‖C‖2 ≥ | logm−O(1)|‖A‖2 for some abso...

in this paper, we considered composition operators on weighted hilbert spaces of analytic functions and  observed that a formula for the  essential norm, gives a hilbert-schmidt characterization and characterizes the membership in schatten-class for these operators. also, closed range composition operators  are investigated.

In these lecture notes, we give an overview about non-local field-theories and their application to polymerized membranes, i.e. membranes with a fixed internal connectivity. The main technical tool is the multi-local operator product expansion (MOPE), generalizing ideas from local field theories to the multi-local situation. These notes are largely inspired by: Kay Wiese, “Polymerized membranes...

In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schrödinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that inverse problem procedure is nothin...

It is known that the classical Hilbert–Schmidt theorem can be generalized to the case of compact operators in Hilbert A-modules H∗ A over a W ∗-algebra of finite type, i.e. compact operators in H∗ A under slight restrictions can be diagonalized over A. We show that if B is a weakly dense C∗-subalgebra of real rank zero in A with some additional property then the natural extension of a compact o...

In Maz'ya (1991), (1994) a new approximation method was proposed mainly directed to the numerical solution of operator equations. This method is characterised by a very accurate approximation in a certain range relevant for numerical computations, but in general the approximations do not converge. For that reason such processes were called approximate approximations (see also Maz'ya and Schmidt...

Lyapunov equation. We analyze the approximation properties of solutions of abstract Lyapunov equations in the setting of a scale of Hilbert spaces associated to an unbounded diagonalizable operator which satisfies the Kato’s square root theorem. We call an (unbounded) operator A diagonalizable if there exists a bounded operator Q, with a bounded inverse, such that the (unbounded) operator Q−1AQ...

Let X and Y be operators on Hilbert space, and let L be a nest of projections on the space. We consider the problem of finding an operator A in Alg L: such that A is Hilbert-Schmidt and such that AX = Y. A necessary and sufficient condition involving X,Y, and the projections in the lattice is found. We also indicate how the statements of the results can be modified so that the main theorem is t...

In this paper we give a constructive characterisation of ultraweakly continuous linear functionals on the space of bounded linear operators on a separable Hilbert space. Let H be a separable complex Hilbert space, with orthonormal basis (en)∞n=1, and B(H) the set of bounded linear operators on H . The weak operator norm associated with the orthonormal basis (en) is defined on B(H) by ‖T ‖w ≡ ∞ ...