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Register a new userRefinements of spectral resolutions and modelling of operators in II$_1$ factors
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Authors
Pedro G. Massey
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We study refinements between spectral resolutions in an arbitrary II$_1$ factor $M$ and obtain diffuse (maximal) refinements of spectral resolutions. We construct models of operators with respect to diffuse spectral resolutions. As an application we obtain new characterizations of sub-majorization and spectral preorder between positive operators in $M$ and n
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Suppose that $phi$ and $psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_phi$ and $T_psi$ denote the Toeplitz operators with symbols $phi$ and $psi$ respectively. We give an explicit formula for the determinant of $T_phi T_psi T_phi^{-1} T_psi^{-1}$ in terms of the products of the tame symbols of $phi$ and $psi$ on the open unit disk.
In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:ell ^{2}(mathbb{N}) to H$, the analysis operator $T^{ast}:Hto $ $% ell ^{2}(mathbb{N}) $ and the associated frame operator $S=TT^{ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{ast}$ and the Gram matrix $G=T^{ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, as well as possible classification of frame sequences with them. For a tight frame sequence, we show that some of these operators are connected in a simple way.
In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_phi$ are orthogonal. When the order $n$ of $phi$ is $2$ or $3$, we show that $M_phi$ is reducible on $D$ if and only if $phi$ is equivalent to $z^n$. When the order of $phi$ is $4$, we determine the reducing subspaces for $M_phi$, and we see that in this case $M_phi$ can be reducible on $D$ when $phi$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $phi$ is not a prime number. Furthermore, we show that $M_phi$ is unitarily equivalent to $M_{z^n} (n > 1)$ on $D$ if and only if $phi = az^n$ for some unimodular constant $a$.
Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a measure space. Suppose that $H$ is a reproducing kernel Hilbert space of functions on $Gtimes Y$, such that $H$ is naturally embedded into $L^2(Gtimes Y)$ and is invariant under the translations associated with the elements of $G$. Under some additional technical assumptions, we study the W*-algebra $mathcal{V}$ of translation-invariant bounded linear operators acting on $H$. First, we decompose $mathcal{V}$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $widehat{H}_xi$, $xiinwidehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $mathcal{V}$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.
Let $A = (a_{j,k})_{j,k=-infty}^infty$ be a bounded linear operator on $l^2(mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-infty}^inftyin l^infty(mathbb{Z})$ are almost periodic sequences. For certain classes of such operators and under certain conditions, we are going to determine the asymptotics of the determinants $det A_{n_1,n_2}$ of the finite sections of the operator $A$ as their size $n_2 - n_1$ tends to infinity. Examples of such operators include block Toeplitz operators and the almost Mathieu operator.
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