65 research outputs found
A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators
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
A hyperbolic universal operator commuting with a compact operator
A 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
On the connected component of compact composition operators on the Hardy space
We show that there exist non-compact composition operators in the connected
component of the compact ones on the classical Hardy space on the unit
disc. This answers a question posed by Shapiro and Sundberg in 1990. We also
establish an improved version of a theorem of MacCluer, giving a lower bound
for the essential norm of a difference of composition operators in terms of the
angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark
measures.Comment: 16 page
Band-diagonal operators on Banach lattices: matrix dynamics and invariant subspaces
We address the existence of non-trivial closed invariant ideals for positive operators defined on Banach lattices whose order is induced by an unconditional basis. In particular, for band-diagonal positive operators such existence is characterized whenever their matrix representations meet a positiveness criteria. For more general classes of positive operators, sufficient conditions are derived proving, particularly, the sharpness of such results from the standpoint of view of the matrix representations. The whole approach is based on studying the behavior of the dynamics of infinite matrices and the localization of the non-zero entries. Finally, we generalize a theorem of Grivaux regarding the existence of non-trivial closed invariant subspaces for positive tridiagonal operators to a more general class of band-diagonal operators showing, in particular, that a large subclass of them have non-trivial closed invariant subspaces but lack non-trivial closed invariant ideals.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEMinisterio de Ciencia e Innovacióninpres
Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy
Finite rank perturbations of diagonalizable normal operators acting boundedly
on infinite dimensional, separable, complex Hilbert spaces are considered from
the standpoint of view of the existence of invariant subspaces. In particular,
if is a rank-one perturbation of a diagonalizable
normal operator with respect to a basis and the vectors and have Fourier coefficients and with respect to respectively, it
is shown that has non trivial closed invariant subspaces provided that
either or have a Fourier coefficient which is zero or and have
non zero Fourier coefficients and
As a consequence, if are such it is shown the existence of non
trivial closed invariant subspaces of whenever
Moreover, such operators have non trivial closed hyperinvariant subspaces
whenever they are not a scalar multiple of the identity. Likewise, analogous
results hold for finite rank perturbations of . This improves
considerably previous theorems of Foia\c{s}, Jung, Ko and Pearcy, Fang and Xia
and the authors on an open question explicitly posed by Pearcy in the
seventies.Comment: Accepted version IUMJ (March 2023
On the Wandering Property in Dirichlet spaces
We show that in a scale of weighted Dirichlet spaces Dα, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in Dα such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when B(z)=zk and |α|≤log(2)log(k+1), the chosen norm is the usual one in Dα
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