Let $$T_n$$ be the linear Hadamard convolution operator acting over Hardy space $$H^q$$ , $$1\le q\le \infty $$ . We call a best approximation-preserving (BAP operator) if $$T_n(e_n)=e_n$$ where $$e_n(z):=z^n,$$ and $$\Vert T_n(f)\Vert _q\le E_n(f)_q$$ for all $$f\in H^q$$ $$E_n(f)_q$$ is approximation by algebraic polynomials of degree most $$n-1$$ in space. give necessary sufficient condition...

A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear mo...

We study the existence of shape-preserving projections and, more generally, the existence of shape-preserving operators with a given ((xed) action.

We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important class of elementary operators with respect to the Schatten p-class. 1. Introduction. Let H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded operators on H into itself. Given A, B ∈ B(H), we define the generalized derivation δ A,B : B(H) B(H) b...

The J-matrix method is extended to difference and q-difference operators and is applied to several explicit differential, difference, q-difference and second order Askey– Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most cases. In some cases we encounter new orthogonal polynom...

This paper completes the proof of the Lp-boundedness of bilinear operators associated to nonsmooth symbols or multipliers begun in Part I, our companion paper [8], by establishing the corresponding Lp-boundedness of time-frequency paraproducts associated with tiles in phase plane. The affine invariant structure of such operators in conjunction with the geometric properties of the associated pha...

In this paper we analyze a class of projection operators with values in a subspace of polynomials. These projection operators are related to the Hubert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthogonality relat...