On the Density of Polynomials in Some Spaces $L^2(M)$

The question of the density of polynomials in some spaces $L^2(M)$ is studied. The following two variants of the measure $M$ and the polynomials are considered: (1) an $N\times N$ matrix-valued nonnegative Borel measure on $\mathbb{R}$ and vector-valued polynomials $p(x)=(p_0(x),p_1(x),\dots,p_{N-1}(x))$, where the $p_j(x)$ are complex polynomials and $N\in \mathbb{N}$; (2) a scalar nonnegative Borel measure on the strip $\Pi=\{(x,\varphi): x\in \mathbb{R}, \, \varphi\in [-\pi,\pi)\}$, and power-trigonometric polynomials $p(x,\varphi)=\sum_{m=0}^\infty\sum_{n=-\infty}^\infty \alpha_{m,n}x^m e^{in\varphi}$, $\alpha_{m,n}\in \mathbb{C}$, where only finitely many $\alpha_{m,n}$ are nonzero. We show that the polynomials are dense in $L^2(M)$ if and only if $M$ is the canonical solution of the corresponding moment problem. It should be stressed that we do not impose any additional constraints on the measure, except the existence of moments. Using the known descriptions of the canonical solutions,, we obtain conditions on the density of polynomials in $L^2(M)$. Simultaneously, we establish a model for commuting self-adjoint and unitary operators with spectrum of finite multiplicity.