Hardy–Hilbert type inequality in reproducing kernel Hilbert space: its applications and related results

ABSTRACT

A reproducing kernel Hilbert space is a Hilbert space $\mathcal{H}=\mathcal{H}\left(\mathrm{\Omega }\right)$ of complex-valued functions on a (non-empty) set Ω, which has the property that point evaluation $f\to f\left(\lambda \right)$ is continuous on $\mathcal{H}$ for all $\lambda \in \mathrm{\Omega }$. Then the Riesz representation theorem guarantees that for every $\lambda \in \mathrm{\Omega }$ there is a unique element $k_{\lambda }=k\left(\cdot ,\lambda \right)\in \mathcal{H}$ such that $f\left(\lambda \right)=⟨f,k_{\lambda }⟩$ for all $f\in \mathcal{H}$. The function $k_{\lambda }$ is called the reproducing kernel of $\mathcal{H}$ and the function ${\hat{k}}_{\lambda }:=k_{\lambda }/{∥k_{\lambda }∥}_{\mathcal{H}}$ is the normalized reproducing kernel in $\mathcal{H}$. The Berezin symbol of an operator A on a reproducing kernel Hilbert space $\mathcal{H}$ is defined by $\tilde{A}\left(\lambda \right)={⟨A{\hat{k}}_{\lambda },{\hat{k}}_{\lambda }⟩}_{\mathcal{H}}.$ The Berezin number of an operator A on $\mathcal{H}$ is defined by $ber\left(A\right)=sup\left\{\left|\tilde{A}\left(\lambda \right)\right|:\lambda \in \mathrm{\Omega }\right\}.$ The so-called Crawford number $c\left(A\right)$ is defined by $c\left(A\right)=inf\left\{\left|⟨Ax,x⟩\right|:x\in \mathcal{H}\mathrm{and}∥x∥=1\right\}.$ We also introduce the number $\tilde{c}\left(A\right)$ defined by $\tilde{c}\left(A\right)=inf\left\{\left|\tilde{A}\left(\lambda \right)\right|:\lambda \in \mathrm{\Omega }\right\}.$ It is clear that $c\left(A\right)\le \tilde{c}\left(A\right)\le ber\left(A\right).$ By using the Hardy–Hilbert type inequality in reproducing kernel Hilbert space, we prove Berezin number inequalities for the convex functions in Reproducing Kernel Hilbert Spaces. We also prove some new inequalities between these numerical characteristics. Some other related results are also obtained.

COMMUNICATED BY:

• Rajesh Pereira

KEYWORDS:

• Numerical radius
• Hardy–Hilbert type inequality
• Berezin number
• self-adjoint operator
• convex function
• unitary operator

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 47A63

Acknowledgments

The authors thank the referee for his useful remarks and suggestions which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by Suleyman Demirel University with Project 4903-YL1-17. Also, the second author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no. RGP-VPP-323.