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| 3. |
- Aleman, Alexandru, et al.
(författare)
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Analytic contractions and boundary behaviour -- an overview.
- 2006
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Ingår i: Proceedings of the first advanced course in operator theory and complex analysis, University of Sevilla, Sevilla, Spain, June 2004.. - 8447210243 ; , s. 3-26
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Konferensbidrag (refereegranskat)
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| 4. |
- Aleman, Alexandru, et al.
(författare)
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Analytic contractions, nontangential limits, and the index of invariant subspaces
- 2007
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Ingår i: Transactions of the American Mathematical Society. - 0002-9947. ; 359:7, s. 3369-3407
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Tidskriftsartikel (refereegranskat)abstract
- Let H be a Hilbert space of analytic functions on the open unit disc D such that the operator M. of multiplication with the identity function. defines a contraction operator. In terms of the reproducing kernel for H we will characterize the largest set Delta(H) subset of partial derivative D such that for each f, g is an element of H, g not equal 0 the meromorphic function f/g has nontangential limits a.e. on Delta( H). We will see that the question of whether or not Delta( H) has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of M-zeta. We further associate with H a second set Sigma(H) subset of partial derivative D, which is defined in terms of the norm on H. For example, Sigma(H) has the property that vertical bar zeta(n)f vertical bar vertical bar -> 0 for all f is an element of H if and only if Sigma( H) has linear Lebesgue measure 0. It turns out that.( H). S( H) a. e., by which we mean that Delta(H) backslash Sigma(H) has linear Lebesgue measure 0. We will study conditions that imply that Delta(H) = Sigma(H) a.e.. As one corollary to our results we will show that if dim H/zeta H = 1 and if there is a c > 0 such that for all f is an element of H and all lambda is an element of D we have parallel to(1-(lambda) over bar zeta)/(zeta-lambda) f parallel to >= c parallel to f||, then Delta(H) = Sigma(H) a.e. and the following four conditions are equivalent: (1) parallel to zeta(n)f parallel to negated right arrow 0 for some f is an element of H, (2) parallel to zeta(n)f parallel to negated right arrow 0 for all f is an element of H, f not equal 0, (3).( H) has nonzero Lebesgue measure, (4) every nonzero invariant subspace M of M-zeta has index 1, i.e., satisfies dim M/zeta M= 1.
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| 5. |
- Aleman, Alexandru, et al.
(författare)
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Derivation-Invariant Subspaces of C∞
- 2008
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Ingår i: Computational Methods in Function Theory. - 1617-9447. ; 8:1-2, s. 493-512
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Tidskriftsartikel (refereegranskat)abstract
- Let $C^\infty(a,b)$ be the Fr\'echet space of all complex-valued infinitely differentiable functions on a (finite or infinite) interval $(a,b)\subset\mathbb{R}$. Let ${L\subset C^\infty(a,b)}$ be a closed subspace such that $DL\subset L$, where $D=\frac{d}{dx}$. Then the spectrum $\sigma_L$ of $D$ on $L$ is either the whole complex plane, or a discrete possibly void set of eigenvalues $\lambda$, each one with some finite multiplicity $m_\lambda\in \mathbb{N}$ such that the monomial exponentials $e_{\lambda,j}(x)=x^j\exp(\lambda x)$, $0\leq j\leq m_\lambda-1$ belong to $L$. If the spectrum is void there is a relatively closed interval $I\subset (a,b)$ such that $L$ consists of those functions from $C^\infty(a,b)$ which vanish identically on $I$. The interval may reduce to a point in which case $L$ consists of the functions that vanish together with all their derivatives at that point.
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| 6. |
- Aleman, Alexandru, et al.
(författare)
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Invariant subspaces for the backward shift on Hilbert spaces of analytic functions with regular norm.
- 2006
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Ingår i: Contemporary Mathematics. - 1098-3627. ; 404, s. 1-25
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that under certain regularity conditions on the norm, the functions in a nontrivial invariant subspace of the backward shift operator have meromorphic pseudocontinuations in the Nevanlinna class of the exterior of the unit disk. In addition, we provide a regularity condition which implies that the subspace itself is contained in the Nevanlinna class of the disc.
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| 7. |
- Aleman, Alexandru, et al.
(författare)
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Nontangential limits in P-t(mu)-spaces and the index of invariant subspaces
- 2009
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Ingår i: Annals of Mathematics. - 0003-486X. ; 169:2, s. 449-490
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Tidskriftsartikel (refereegranskat)abstract
- Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P-t(mu) whenever P-t(mu) not equal L-t(mu) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure nu in the complex plane we are able to describe locations of bounded point evaluations for P-t(nu) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < infinity dim M/zM = 1 for every nonzero invariant subspace M of P-t(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in P-t(mu) near the points z is an element of partial derivative D where h(z) = 0. In particular, for 1 < t < infinity we show that there are interpolating sequences for P-t(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}.
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| 8. |
- Aleman, Alexandru, et al.
(författare)
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Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups
- 2009
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Ingår i: Annals of Mathematics. - 0003-486X. ; 169:2, s. 449-490
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Tidskriftsartikel (refereegranskat)abstract
- Abstract Let μ be a finite positive measure on the closed disk D¯ in the complex plane, let 1 ≤ t < ∞, and let Pt(μ) denote the closure of the analytic polynomials in Lt(μ). We suppose that D is the set of analytic bounded point evaluations for Pt(μ), and that Pt(μ) contains no nontrivial characteristic functions. It is then known that the restriction of μ to ∂D must be of the form h|dz|. We prove that every function f ∈ Pt(μ) has nontangential limits at h|dz|-almost every point of ∂D, and the resulting boundary function agrees with f as an element of Lt(h|dz|). Our proof combines methods from James E. Thomson’s proof of the existence of bounded point evaluations for Pt(μ) whenever Pt(μ)≠Lt(μ) with Xavier Tolsa’s remarkable recent results on analytic capacity. These methods allow us to refine Thomson’s results somewhat. In fact, for a general compactly supported measure ν in the complex plane we are able to describe locations of bounded point evaluations for Pt(ν) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < ∞ dim ℳ∕zℳ = 1 for every nonzero invariant subspace ℳ of Pt(μ) if and only if h≠0. We also investigate the boundary behaviour of the functions in Pt(μ) near the points z ∈ ∂D where h(z) = 0. In particular, for 1 < t < ∞ we show that there are interpolating sequences for Pt(μ) that accumulate nontangentially almost everywhere on {z : h(z) = 0}.
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| 9. |
- Aleman, Alexandru, et al.
(författare)
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Preduals of $Q_p$-spaces.
- 2007
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Ingår i: Complex Variables and Elliptic Equations. - 1747-6933. ; 52:7, s. 605-628
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Tidskriftsartikel (refereegranskat)abstract
- Summary: We prove a weak factorization result for the predual of the space $Q_p$ on the unit disc, that is, we show that functions in this space can be written as sums of products of functions in given weighted Dirichlet and Bergman spaces with the usual control on the norms.
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| 10. |
- Aleman, Alexandru, et al.
(författare)
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Preduals of $Q_p$-spaces. II: Carleson imbeddings and atomic decompositions
- 2007
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Ingår i: Complex Variables and Elliptic Equations. - 1747-6933. ; 52:7, s. 629-653
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Tidskriftsartikel (refereegranskat)abstract
- Summary: In [Aleman, A., Carlsson, M. and Persson A., Preduals of $Q_p$-spaces. Complex Variables (To appear).], we have obtained a representation of the Cauchy-predual of the space $Q_p$ on the unit disc as a weak product of weighted Dirichlet and Bergman spaces. The present article is a continuation of Aleman et al. and contains several applications and further developments of those results. We investigate the relation between $Q_p$ and Carleson inequalities for functions in weighted Dirichlet spaces and, in particular, we prove a characterization of bounded $Q_p$-functions in terms of pointwise multipliers between such spaces. Moreover, we use our approach based on imbeddings in vector-valued sequence spaces to obtain atomic decompositions of the predual of $Q_p$. This last result is then extended to the real variable setting where we prove atomic decomposition theorems for the preduals of certain function spaces that generalize $Q_p(mathbb R^n)$.
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