Essential norm of generalized Hilbert matrix from Bloch type spaces to BMOA and Bloch space

TL;DR

This paper characterizes the boundedness, compactness, and essential norm of Hankel matrices induced by positive measures when acting between Bloch type spaces, BMOA, and Bloch space, advancing understanding of operator behavior in complex analysis.

Contribution

It provides new characterizations of boundedness, compactness, and computes the essential norm of Hankel operators between Bloch type spaces, BMOA, and Bloch space.

Findings

Characterization of boundedness of Hankel operators between spaces.

Criteria for compactness of these operators.

Explicit formulas for the essential norm.

Abstract

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_\mu=(\mu_{n+k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$ induces the operator $$ \mathcal{H}_\mu(f)(z)=\sum^\infty_{n=0}\left(\sum^\infty_{k=0}\mu_{n,k}a_k\right)z^n $$ on the space of all analytic functions $f(z)=\sum^\infty_{n=0}a_nz^n$ in the unit disk $\mathbb{D}$. In this paper, we characterize the boundedness and compactness of $\mathcal{H}_\mu$ from Bloch type spaces to the BMOA and the Bloch space. Moreover we obtain the essential norm of $\mathcal{H}_\mu$ from $\alpha$ Bloch type spaces to Bloch space and BMOA.