Refined Inequalities For The q -Berezin Radii Of Operators On Reproducing Kernel Hilbert Spaces

Let [Formula: see text] be a reproducing kernel Hilbert space, fix [Formula: see text], and assume that its normalized kernel family contains pairs with inner product [Formula: see text]. We obtain upper bounds for the [Formula: see text]-Berezin radius in terms of the Berezin number and [Formula: see text], where [Formula: see text] denotes the kernel norm. Applying Buzano’s inequality and a Gram determinant inequality to [Formula: see text] and [Formula: see text] gives further bounds involving [Formula: see text] and [Formula: see text], with [Formula: see text]. We also obtain inequalities for [Formula: see text] operator matrices using numerical-radius estimates and a specified family of normalized kernels. Applications include operator products and commutators. A finite-dimensional example compares the bounds with an earlier estimate. A finite-rank operator on the Hardy space gives equality in the Buzano bound at [Formula: see text].