Browsing by Author "Cowen, Carl C."
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Item Commutants of composition operators on the Hardy space of the disk(2013-11-06) Carter, James Michael; Cowen, Carl C.; Klimek, Slawomir; Perez, Rodrigo A.; Chin, Raymond; Bell, Steven R.; Mukhin, EvgenyThe main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.Item Constructing invariant subspaces as kernels of commuting matrices(Elsevier, 2019-12) Cowen, Carl C.; Johnston, William; Wahl, Rebecca G.; Mathematical Sciences, School of ScienceGiven an n n matrix A over C and an invariant subspace N, a straightforward formula constructs an n n matrix N that commutes with A and has N = kerN. For Q a matrix putting A into Jordan canonical form, J = Q1AQ, we get N = Q1M where M= ker(M) is an invariant subspace for J with M commuting with J. In the formula J = PZT1Pt, the matrices Z and T are m m and P is an n m row selection matrix. If N is a marked subspace, m = n and Z is an n n block diagonal matrix, and if N is not a marked subspace, then m > n and Z is an m m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a nite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A.Item Convexity of the Berezin Range(Elsevier, 2022) Cowen, Carl C.; Felder, Christopher; Mathematical Sciences, School of ScienceThis paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B(T):={〈Tkˆx,kˆx〉H:x∈X}, where kˆx is the normalized reproducing kernel for H at x∈X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.Item D-bar and Dirac Type Operators on Classical and Quantum Domains(2012-08-29) McBride, Matthew Scott; Klimek, Slawomir; Cowen, Carl C.; Ji, Ron; Dadarlat, MariusI study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.Item A hyperbolic universal operator commuting with a compact operator(AMS, 2019) Cowen, Carl C.; Gallardo-Gutiérrez, Eva A.; Mathematical Sciences, School of ScienceA Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a non-trivial, quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a `hyperbolic' composition operator.Item A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators(American Mathematical Society, 2017) Cowen, Carl C.; Gallardo-Gutiérrez, Eva A.A striking result by Nordgren, Rosenthal and Wintrobe states that the Invariant Subspace Problem is equivalent to the fact that any minimal invariant subspace for a composition operator Cφ induced by a hyperbolic automorphism φ of the unit disc D acting on the classical Hardy space H² is one dimensional. We provide a completely different proof of Nordgren, Rosenthal and Wintrobe’s Theorem based on analytic Toeplitz operators.Item Restrictions to Invariant Subspaces of Composition Operators on the Hardy Space of the Disk(2014-01-29) Thompson, Derek Allen; Cowen, Carl C.; Ji, Ronghui ; Klimek, Slawomir; Bell, Steven R.; Mukhin, EvgenyInvariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.Item Rota's universal operators and invariant subspaces in Hilbert spaces(Elsevier, 2016-09) Cowen, Carl C.; Gallardo-Gutiérrez, Eva A.; Department of Mathematical Sciences, School of ScienceA Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.