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Register a new userNumerical Range and Compressions of the Shift
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The numerical range of a bounded, linear operator on a Hilbert space is a set in $mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 ominus theta H^2$ where $theta$ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture.
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We consider two-variable model spaces associated to rational inner functions $Theta$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $mathcal{S}_2$, namely $ S^1_{Theta}:= P_{mathcal{S}_2} M_{z_1} |_{mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and study the geometry of the associated numerical ranges, find formulas for the boundaries, answer the zero inclusion question, and determine whether the numerical ranges are ever circular.
Let $mathfrak{n}$ be a nonempty, proper, convex subset of $mathbb{C}$. The $mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.
A result by Liskevich and Perelmuter from 1995 yields the optimal angle of analyticity for symmetric submarkovian semigroups on $L_p$, $1<p<infty$. C.~Kriegler showed in 2011 that the result remains true without the assumption of positivity of the semigroup. Here we give an elementary proof of Krieglers result.
We obtain sufficient conditions for a densely defined operator on the Fock space to be bounded or compact. Under the boundedness condition we then characterize the compactness of the operator in terms of its Berezin transform.
The corona problem was motivated by the question of the density of the open unit disk D in the maximal ideal space of the algebra, H1(D), of bounded holomorphic functions on D. In this note we study relationships of the problem with questions in operator theory and complex geometry. We use the framework of Hilbert modules focusing on reproducing kernel Hilbert spaces of holomorphic functions on a domain, in Cm. We interpret several of the approaches to the corona problem from this point of view. A few new observations are made along the way. 2012 MSC: 46515, 32A36, 32A70, 30H80, 30H10, 32A65, 32A35, 32A38 Keywords: corona problem, Hilbert modules, reproducing kernel Hilbert space, commutant lifting theorem 1
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