# On sum of two subnormal kernels

## TL;DR

This paper demonstrates through examples that the sum of two subnormal kernels on the unit disc does not necessarily produce a subnormal multiplication operator, disproving a recent conjecture and exploring cases where it does.

## Contribution

It provides counterexamples to a conjecture about subnormal kernels and discusses conditions under which the sum remains subnormal.

## Key findings

- Counterexamples show sum of subnormal kernels need not be subnormal.
- The conjecture by Adams, Feldman, and McGuire is false in general.
- Certain cases where the sum is subnormal are identified.

## Abstract

We show, by means of a class of examples, that if $K_1$ and $K_2$ are two positive definite kernels on the unit disc such that the multiplication by the coordinate function on the corresponding reproducing kernel Hilbert space is subnormal, then the multiplication operator on the Hilbert space determined by their sum $K_1+K_2$ need not be subnormal. This settles a recent conjecture of Gregory T. Adams, Nathan S. Feldman and Paul J. McGuire in the negative. We also discuss some cases for which the answer is affirmative.