For a large class of admissible functions f : R 7→ C, the operator derivatives dj dxj f(H0 + xV ), where H0 and V are self-adjoint operators on a separable Hilbert space H, exist and can be represented as multiple operator integrals [1, 14]. Let M be a semi-finite von Neumann algebra acting on H and τ a semi-finite normal faithful trace on M. For H0 = H ∗ 0 affiliated with M and V = V ∗ in the ...

We study the most general conditions under which the computations of the index of a perturbed Dirac operator Ds = D + sZ localize to the singular set of the zeroth order operator Z in the semi-classical limit s → ∞. We use Witten’s method to compute the index of D by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator Z. The paper contains...

We study the most general conditions under which the computations of the index of a perturbed Dirac operator Ds = D + sZ localize to the singular set of the zeroth order operator Z in the semi-classical limit s → ∞. We use Witten’s method to compute the index of D by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator Z. The paper contains...

Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is consider...

for every rational function ƒ with poles off K. In this note it is shown that any operator for which the spectrum is a spectral set has a nontrivial invariant subspace. In [6] von Neumann introduced the notion of spectral set and showed that if T has II T\\ = 1 then the closed unit disc, D"~, is a spectral set for T. For this reason any operator whose spectrum is a spectral set is called a von ...

We consider Khudaverdian's geometric version of a Batalin-Vilkovisky (BV) operator ∆ E in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semi-densities to semidensities. We find a local formula for the ∆ E operator in arbitrary coordinates. As an imp...

We consider Khudaverdian's geometric version of a Batalin-Vilkovisky (BV) operator ∆ E in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semi-densities to semidensities. We find a local formula for the ∆ E operator in arbitrary coordinates. As an imp...

We consider Khudaverdian's geometric version of a Batalin-Vilkovisky (BV) operator ∆ E in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semi-densities to semidensities. We find a local formula for the ∆ E operator in arbitrary coordinates. As an imp...

Let (H ; 〈·, ·〉) be a complex Hilbert space and T : H → H a bounded linear operator on H. Recall that T is a normal operator if T T = TT . Normal operators may be regarded as a generalisation of self-adjoint operator T in which T ∗ need not be exactly T but commutes with T [11, p. 15]. The numerical range of an operator T is the subset of the complex numbers C given by [11, p. 1]: W (T ) = {〈Tx...

We introduce the class of quasi-square-2-isometric operators on a complex separable Hilbert space. This extends 2-isometric due to Agler and Stankus. An operator T is said be if T*5T5 ? 2T*3T3 + T*T = 0. In this paper, we give matrix representation in order obtain spectral properties operator. particular, show that function continuous all operators. Under hypothesis ?(T)?(??(T)) ?, also prove E...