New Inequalities and an Integral Expression for the 𝒜‐Berezin Number

Source abstract

This work examines a reproducing kernel Hilbert space constructed on a nonempty set . Our investigation focuses on the ‐Berezin number and the ‐Berezin norm, where denotes a positive bounded linear operator acting on . For an ‐bounded linear operator , the ‐Berezin seminorm is defined by where and are normalized reproducing kernels in , and for all . Similarly, the ‐Berezin number is given by The primary aim of this paper is to establish several new inequalities involving the ‐Berezin seminorm and the ‐Berezin number. Furthermore, we derive an explicit integral expression for the ‐Berezin number and establish a necessary and sufficient condition for equality in the associated triangle inequality.