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This paper is a continuation of our previous investigations on the truncatedmatrix trigonometric moment problem in Ukrainian Math. J., 2011, \textbf{63},no. 6, 786-797, and Ukrainian Math. J., 2013, \textbf{64}, no. 8, 1199-1214. Inthis paper we shall study the truncated matrix trigonometric moment problemwith an additional constraint posed on the matrix measure$M_{\mathbb{T}}(\delta)$, $\delta\in \mathfrak{B}(\mathbb{T})$, generated bythe seeked function $M(x)$: $M_{\mathbb{T}}(\Delta) = 0$, where $\Delta$ is agiven open subset of $\mathbb{T}$ (called a gap). We present necessary andsufficient conditions for the solvability of the moment problem with a gap. Allsolutions of the moment problem with a gap can be constructed by aNevanlinna-type formula.

The truncated matrix trigonometric moment problem with an open gap