Characterization of Orthogonal Polynomials in Norm-Attainable Classes

Abstract

Orthogonal polynomials within the realm of bounded linear operators on a Hilbert space (B(H )) hold a crucial role in operator theory, functional analysis, and various other fields. While significant research has been conducted on orthogonal polynomials and norm-attainable operators, there remains a notable gap in the literature regarding the characterization of orthogonal polynomials in (N A(H )) and the relationship between orthog- onal polynomials and norm-attainable operators. This study aims to ad- dress this knowledge gap by characterizing norm-attainable operators, or- thogonal polynomials in N A(H ), and establishing their relationship. The research utilizes diverse methodologies such as norm-attainability cri- teria, Gram-Schmidt orthonormalization, determination of Gram ma- trix determinants, and exploration of properties associated with classi- cal continuous orthogonal polynomials. The results of the study demon- strate that Harmite, Laguerre, Legendre, and Jacobi polynomials are norm-attainable in B(H ), and a Hermitian contraction operator is norm- attainable if its ∥T ∥ or −∥T ∥ norm lies within its spectrum. Furthermore, it is revealed that orthogonal polynomials exhibit convexity, positivity, and form a normed vector space. Additionally, a self-adjoint closed dif- ferential operator on the L2([0, 1]) space, which is not bounded and hence not norm-attainable, is identified. This study holds significance as it enhances our comprehension of norm-attainable operators and orthogonal polynomials within the B(H ) space. The findings have practical applications in signal processing, data analysis, and harmonic analysis, particularly in the development of Fourier series, wavelength determination, and the L2-boundedness of singular integral operators.