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GRAMIAN ORTHOGONALLY SCATTERED DILATION OF $B(\mathcal{U}, \mathcal{H})$-VALUED MEASURES WITH APPLICATIONS
Keywords:
gramian orthogonally scattered measures, gramian orthogonally scattered dilation, operator stationary processes, operator stationary dilation.DOI:
https://doi.org/10.17654/0972087123011Abstract
For a Banach space $\mathcal{U}$ and a Hilbert space $\mathcal{H}, B(\mathcal{U}, \mathcal{H})$-valued measures are studied. $B(\mathcal{U}, \mathcal{H})$ is a right $B(\mathcal{U})$-module and has a $B\left(\mathcal{U}, \mathcal{U}^*\right)$-valued gramian. Gramian orthogonally scattered measures are defined and some necessary and sufficient conditions are obtained for a $B(\mathcal{U}, \mathcal{H})$-valued measure to have a gramian orthogonally scattered dilation. In particular, if $\mathcal{U}$ has a Schauder basis, then some more conditions can be shown. These are applied to consider operator stationary dilation of $B(\mathcal{U}, \mathcal{H})$-valued harmonizable processes on a locally compact abelian group.
Received: April 6, 2023
Accepted: May 11, 2023
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