Kowalski–Słodkowski Theorem for Reproducing Kernel Hilbert Spaces

On reproducing kernel Hilbert spaces with normalized complete Pick kernel, we establish an equivalent result to the Gleason–Kahane–Żelazko theorem without assuming linearity. On the way of establishing this, we observe that linearity on a multiplier algebra is enough to conclude linearity on the whole Hilbert space. By constructing a counter-example, we show that the condition of complete Pick kernel can not be removed. Also, we demonstrate the automatic continuity of such functionals. Leveraging these findings, we extend the Kowalski–Słodkowski theorem in this setup.