This contribution deals with multiple orthogonal polynomials of type II with respect to q-discrete measures (q-Hahn measures). In addition, we show that this family of multiple orthogonal polynomials has a lowering operator, and raising operators as well as a Rodrigues type formula. The combination of lowering and raising operators leads to a third order q-difference equation when two orthogona...

This paper deals with the study of Birkhoff--James orthogonality a linear operator to subspace operators defined between arbitrary Banach spaces. In case domain space is reflexive and finite dimensional we obtain complete characterization. For spaces, same under some additional conditions. an Hilbert H, also L(H), both respect norm as well numerical radius norm.

Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given positive operator $A\in\B(\h)$, and number $\lambda\in [0,1]$, seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined set $\B_{A^{1/2}}(\h)$ in $\B(\h)$ having an $A^{1/2}$-adjoint. The combination sesquilinear form ${\langle \cdot, \...

We consider the product of spectral projections Πε(λ) = 1(−∞,λ−ε)(H0)1(λ+ε,∞)(H)1(−∞,λ−ε)(H0) where H0 and H are the free and the perturbed Schrödinger operators with a short range potential, λ > 0 is fixed and ε → 0. We compute the leading term of the asymptotics of Tr f(Πε(λ)) as ε → 0 for continuous functions f vanishing sufficiently fast near zero. Our construction elucidates calculations t...

We investigate dispersive estimates for the Schrödinger operator H = −∆+V with V is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion eχ(H)Pac(H) = O(1/(log t))A0 +O(1/t)A1 +O((t log t) )A2 +O(t (log t))A3. Here A0, A1 : L (R) → L∞(Rn), while A2, A3 are o...

We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results Koldobsky and Wójcik. Moreover, in an arbitrary dimension, we construct examples non-isometric pairs non-smooth admit norm preserving homogeneous bicontinuous prese...

The Pólya-Schur theory describes the class of hyperbolicity preservers, i.e., the linear operators on univariate polynomials preserving realrootedness. We attempt to develop an analog of Pólya-Schur theory in the setting of linear finite difference operators. We study the class of linear finite difference operators preserving the set of real-rooted polynomials whose mesh (i.e., the minimal dist...