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Maria F. Gamal'

Publications and source records attributed to Maria F. Gamal'.

15 recordsLinked to original sources
arXiv2026

Non-stable subnormal contractions have nontrivial hyperinvariant subspaces

A contraction $T$ on a (complex, separable) Hilbert space is stable, or of class $C_{0\cdot}$, if $T^n\to 0$ in the strong operator topology. It is proved that for a non-stable pure subnormal contraction $T$ there exists a singular inner function $θ$ such that the range of $θ(T)$ is not dense. Consequently, $T$ has nontrivial hyperinvariant subspaces. The proof is based on results by Esterle and Kérchy. Examples of stable subnormal contractions are given for which the range of $φ(T)$ is dense for every $φ\in H^\infty$ ($φ\not\equiv 0$).

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arXiv2025

Some relationships with subnormal operators and existence of hyperinvariant subspaces

If $T$ is a polynomially bounded operator, $\mathcal M$ is an invariant subspace of $T$, $T|_{\mathcal M}$ is a unilateral shift and $T^*|_{\mathcal M^\perp}$ is subnormal, then $T$ has a nontrivial hyperinvariant subspace. If an operator $T$ is intertwined from both sides with two operators, one of which is hyponormal and other is the adjoint to hyponormal, then $T$ has a nontrivial hyperinvariant subspace. The existence of nontrivial hyperinvariant subspaces for subnormal operators themselves is not studied here.

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arXiv2024

Examples of cyclic polynomially bounded operators that are not similar to contractions

A question if a polynomially bounded operator is similar to a contraction was posed by Halmos and was answered in the negative by Pisier. His counterexample is an operator of infinite multiplicity, while all its restrictions on invariant subspaces of finite multiplicity are similar to contractions. In the paper, cyclic polynomially bounded operators which are not similar to contractions and are quasisimilar to $C_0$-contractions or to isometries are constructed. The construction is based on a perturbation of the sequence of finite dimensional operators which is uniformly polynomially bounded, but is not uniformly completely polynomially bounded, constructed by Pisier.

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arXiv2024

On similarity to contractions of class $C_{\cdot 0}$ with finite defects

A criterion on the similarity of a (bounded, linear) operator $T$ on a (complex, separable) Hilbert space $\mathcal H$ in terms of shift-type invariant subspaces of $T$ to a contraction of class $C_{\cdot 0}$ with finite unequal defects is given. Namely, $T$ is similar to such a contraction if and only if the minimal quantity of (closed) invariant subspaces $\mathcal M$ of $T$ such that the restriction $T|_{\mathcal M}$ of $T$ on $\mathcal M$ is similar to the simple unilateral shift, whose linear span is $\mathcal H$, is finite. A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator $T$ to a contraction of class $C_{\cdot 0}$ with finite equal defects is given. Namely, $T$ is similar to such a contraction if the (spectral) multiplicity of $T$ is finite and $B(T)=\mathbb O$, where $B$ is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson--Newman product).

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arXiv2023

On expansive operators that are quasisimilar to the unilateral shift of finite multiplicity

An operator $T$ on a Hilbert space $\mathcal H$ is called expansive, if $\|Tx\|\geq \|x\|$ ($x\in\mathcal H$). Expansive operators $T$ quasisimilar to the unilateral shift $S_N$ of finite multiplicity $N$ are studied. It is proved that $I-T^*T$ is of trace class for such $T$. Also the lattice $\mathrm{Lat}T$ of invariant subspaces of an expansive operator $T$ quasisimilar to $S_N$ is studied. It is proved that $\dim\mathcal M\ominus T\mathcal M\leq N$ for every $\mathcal M\in\mathrm{Lat}T$. It is shown that if $N\geq 2$, then there exist $\mathcal M_j\in\mathrm{Lat}T$ ($j=1,\ldots, N$) such that the restriction $T|_{\mathcal M_j}$ of $T$ on $\mathcal M_j$ is similar to the unilateral shift $S$ of multiplicity $1$ for every $j=1,\ldots, N$, and $\mathcal H=\vee_{j=1}^N\mathcal M_j$. For $N=1$, that is, for $T$ quasisimilar to $S$, there exist two spaces $\mathcal M_1$, $\mathcal M_2\in\mathrm{Lat}T$ such that $T|_{\mathcal M_j}$ is similar to $S$ for $j=1,2$, and $\mathcal H=\mathcal M_1\vee\mathcal M_2$. Example of an expansive operator $T$ quasisimilar to $S$ is given such that intertwining transformations do not give an isomorphism of $\mathrm{Lat}T$ and $\mathrm{Lat}S$.

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arXiv2022

On isometric asymptotes of operators quasisimilar to isometries

The notion of isometric and unitary asymptotes was introduced for power bounded operators in 1989 and was generalized in 2016--2019 by Kérchy. In particular, it was shown that there exist operators without unitary asymptote. In this paper operators are constructed which are quasisimilar to isometries and do not have isometric asymptotes. Also a contraction is constructed which is quasisimilar to the unilateral shift of infinite multiplicity and whose isometric asymptote contains a (non-zero) unitary summand.

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arXiv2019

On polynomially bounded operators with shift-type invariant subspaces

A particular case of [07] was generalized from contractions to polynomially bounded operators in [G19]. Namely, it is proved in [G19] that if the unitary asymptote of a polynomially bounded operator $T$ contains the bilateral shift of multiplicity $1$, then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of multiplicity $1$. In the present paper, some corollaries of this result are given. In particular, reflexivity of polynomially bounded operators described above is proved.

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arXiv2019

On existence of shift-type invariant subspaces for polynomially bounded operator

A particular case of results from [K2] is as follows. Let the unitary asymptote of a contraction $T$ contain the bilateral shift (of finite or infinite multiplicity). Then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of the same multiplicity. The proof is based on the Sz.-Nagy--Foias functional model for contractions. In the present paper this result is generalized to polynomially bounded operators, but in the simplest case. Namely, it is proved that if the unitary asymptote of a polynomially bounded operator $T$ contains the bilateral shift of multiplicity $1$, then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of multiplicity $1$. The proof is based on a result from [B].

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arXiv2019

On power bounded operators with holomorphic eigenvectors, II

In [U] (among other results), M. Uchiyama gave the necessary and sufficient conditions for contractions to be similar to the unilateral shift $S$ of multiplicity $1$ in terms of norm-estimates of complete analytic families of eigenvectors of their adjoints. In [G2], it was shown that this result for contractions can't be extended to power bounded operators. Namely, a cyclic power bounded operator was constructed which has the requested norm-estimates, is a quasiaffine transform of $S$, but is not quasisimilar to $S$. In this paper, it is shown that the additional assumption on a power bounded operator to be quasisimilar to $S$ (with the requested norm-estimates) does not imply similarity to $S$. A question whether the criterion for contractions to be similar to $S$ can be generalized to polynomially bounded operators remains open. Also, for every cardinal number $2\leq N\leq \infty$ a power bounded operator $T$ is constructed such that $T$ is a quasiaffine transform of $S$ and $\dim\ker T^*=N$. This is impossible for polynomially bounded operators. Moreover, the constructed operators $T$ have the requested norm-estimates of complete analytic families of eigenvectors of $T^*$.

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arXiv2018

Some sufficient conditions for existence of hyperinvariant subspaces for operators intertwined with unitaries

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of $φ(T)$. Here $φ$ is a singular inner function, if $T$ is polynomially bounded, or $φ$ is an analytic in the unit disc function with absolutely summable Taylor coefficients and singular inner part, if $T$ is supposed to be power bounded only. Also, an example of a quasianalytic contraction $T$ is given. The quasianalytic spectral set of $T$ is not the whole unit circle $\mathbb T$, while $σ(T)=\mathbb T$. Proofs are based on results by Esterle, Kellay, Borichev and Volberg.

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arXiv2018

On power bounded operators that are quasiaffine tranforms of singular unitaries

In [9] a question is raised: if a power bounded operator is quasisimilar to a singular unitary operator, is it similar to this unitary operator? For polynomially bounded operators, a positive answer to this question is known [1], [13]. In this paper a positive answer is given in some particular cases, but in general an answer remains unknown.

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arXiv2017

One dimensional perturbations of unitaries that are quasiaffine transforms of singular unitaries, and multipliers between model spaces

It is shown that, under some natural additional conditions, an operator which intertwines one cyclic singular unitary operator with one dimensional perturbation of another cyclic singular unitary operator is the operator of multiplication by a multiplier between model spaces. Using this result, it is shown that if $T$ is one dimensional perturbation of a unitary operator, $T$ is a quasiaffine transform of a singular unitary operator, and $T$ is power bounded, then $T$ is similar to a unitary operator, and $\sup_{n\geq 0}\|T^{-n}\|\leq(2(\sup_{n\geq 0}\|T^n\|)^2+1)\cdot(\sup_{n\geq 0}\|T^n\|)^5$.

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