On the spectrum of the symmetric tensor products of certain Hilbert-space operators

This paper primarily investigates spectral properties of symmetric tensor products of Hilbert-space operators. For a unilateral weighted shift operator $S_w$, we present an algorithm to compute the point spectrum of its symmetric and antisymmetric tensor products with the adjoint $S_w^{\ast }$. Additionally, we analyze the symmetric tensor product of an injective unilateral weighted shift $S_{\alpha }$ and a diagonal operator $M$ on $l^2$, demonstrating that its point spectrum must be contained in $\{0\}$.