Motivated by the study of multiplicative linear functionals in reproducing kernel Hilbert space (RKHS) with normalized complete Pick kernel, we define and study the multiplicative linear map between two RKHS. We identify the conditions under which such maps are continuous. Additionally, we prove that any unital cyclicity-preserving linear map is multiplicative. Conversely, we also characterize when a multiplicative linear map is unital cyclicity preserving. These results serve as a generalization of the Gleason–Kahane–Żelazko theorem to the setting of multiplicative maps between two RKHS. We present the composition operator as a natural class of examples of multiplicative linear maps on an RKHS. We also prove that every continuous multiplicative linear operator can be realized as a composition operator on various analytic Hilbert spaces over the unit disc $\mathbb {D}.$

We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a $C^\star $-algebra, and if $\phi :A\to \mathbb C $ is a continuous function satisfying $ \phi (x)\phi (y) \in \sigma (xy) $ for all $x,y\in A$ (where $\sigma $ denotes the spectrum), then either $\phi $ is a character of A or $-\phi $ is a character of A.

If $A$ is a commutative $C^{\star }$-algebra and if $\unicode[STIX]{x1D719}:A\rightarrow \mathbb{C}$ is a continuous multiplicative functional such that $\unicode[STIX]{x1D719}(x)$ belongs to the spectrum of $x$ for each $x\in A$, then $\unicode[STIX]{x1D719}$ is linear and hence a character of $A$. This establishes a multiplicative Gleason–Kahane–Żelazko theorem for $C(X)$.