Regularized trace on separable Banach spaces

If H is a separable Hilbert space, Gül (2008) has shown that a regularized trace formula can be computed on L² (H, [0, π]) for a second order differential operator with bounded operator-valued coefficients, where H is a separable Hilbert space. Kuelbs (1970) has shown that every separable Banach space can be continuously and densely embedded into a separable Hilbert space, while Gill (2016) has used Kuelbs result to show that the dual of a Banach space does not have a unique representation. In this paper, we use the results of Kuelbs and Gill to study the regularized trace formula on L2 (B, [0, π]), where B is an arbitrary separable Banach space.Publisher's Versio

Trace Formulas for a Conformable Fractional Diffusion Operator

In this paper, the regularized trace formulas for a diffusion operator which include conformable fractional derivatives of order {\alpha} (0<{\alpha \leq 1}) is obtained.Comment: 12 page

Spectral inequalities for discrete and continuous differential operators

In this thesis spectral inequalities and trace formulae for discrete and continuous differential operators are discussed. We first investigate spectral inequalities for Jacobi operators with matrix-valued potentials and present a new, direct proof of a sharp inequality corresponding to a Lieb–Thirring inequality for the power 3/2 using the commutation method. For the special case of a discrete Schrödinger operator we also prove new inequalities for higher powers of the eigenvalues and the potential and compare our results to previously established bounds. We then approximate a Schrödinger operator on L^2(\R) by Jacobi operators on \ell^2(\Z) and use the established inequalities to provide new proofs of sharp Lieb–Thirring inequalities for the powers \gamma=1/2 and \gamma=3/2. By means of interpolation we derive spectral inequalities for Jacobi operators that yield (non-sharp) Lieb–Thirring constants on the real line for powers 1/2<\gamma<3/2. We then consider Schrödinger operators on a finite interval [0,b] with matrix-valued potentials and establish trace formulae of the Buslaev–Faddeev–Zakharov type. The results link sums of powers of the negative eigenvalues to terms dependent on the potential and scattering functions. Finally, we discuss the Berezin inequality, which is well-known on sets of finite measure and find an analogous inequality for the magnetic operator with constant magnetic field on a set whose complement has finite measure. We obtain a similar bound for the Heisenberg sub-Laplacian.Open Acces