| 1. |
- Nedic, Mitja, 1990-
(författare)
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Integral representations of Herglotz-Nevanlinna functions
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis, we study integral representations of Herglotz-Nevanlinna functions, that is to say holomorphic functions defined on a product of several copies of the complex upper half-plane having non-negative imaginary part. The manuscript is divided into three parts, beginning with a general introduction followed by two papers.In the general introduction, we familiarize ourselves with the concept of a Herglotz-Nevanlinna function as well as providing a comprehensive introduction into the theory of integral representations for this particular class of functions.Paper I treats exclusively the two-variable case and presents an integral representation of Herglotz-Nevanlinna functions in two complex variables in terms of a real number, two non-negative numbers and a positive Borel measure satisfying two properties. Three properties that hold for the class of measures appearing in such integral representations are also proven.In Paper II, we provide an integral representation for the class of Herglotz-Nevanlinna functions in arbitrarily many complex variables in terms of a real number, a linear term and a positive Borel measure satisfying two properties. Properties of the class of measures appearing in this representation are then discussed in detail as well as alternative descriptions of said class. Finally, a symmetry formula satisfied by Herglotz-Nevanlinna functions is proved at the end.
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| 2. |
- Aleman, Alexandru, et al.
(författare)
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Density of disk algebra functions in de Branges–Rovnyak spaces
- 2017
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Ingår i: Comptes rendus. Mathematique. - : Elsevier BV. - 1631-073X .- 1778-3569. ; 355:8, s. 871-875
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Tidskriftsartikel (refereegranskat)abstract
- We prove that functions analytic in the unit disk and continuous up to the boundary are dense in the de Branges–Rovnyak spaces induced by the extreme points of the unit ball of . Together with previous theorems, it follows that this class of functions is dense in any de Branges–Rovnyak space.
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| 3. |
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| 4. |
- Aleman, Alexandru, et al.
(författare)
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A Quantitative Estimate for Bounded Point Evaluations in P-t(mu)-spaces
- 2010
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Ingår i: Topics In Operator Theory: Operators, Matrices And Analytic Functions, Vol 1. - 0255-0156. ; 202, s. 1-10
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Konferensbidrag (refereegranskat)abstract
- In this note we explain how X. Tolsa's work on analytic capacity and an adaptation of Thomson's coloring scheme can be used to obtain a quantitative version of J. Thomson's theorem on bounded point evaluations for P-t(mu)-spaces.
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| 5. |
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| 6. |
- 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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| 7. |
- 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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| 8. |
- Aleman, Alexandru, et al.
(författare)
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Beurling's theorem for the Bergman space
- 1996
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Ingår i: Acta Mathematica. - 0001-5962. ; 177:2, s. 275-310
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Tidskriftsartikel (refereegranskat)abstract
- A celebrated theorem in operator theory is A. Beurling's description of the invariant subspaces in $H^2$ in terms of inner functions [Acta Math. {\bf81} (1949), 239--255; MR0027954 (10,381e)]. To do the same thing for the Bergman space $L^2_a$ has been deemed virtually impossible by many analysts, in view of the fact that the lattice of invariant subspaces is so large, and that the invariant subspaces may have weird properties as viewed from the $H^2$ perspective. The size of the lattice can be appreciated from the known fact that essentially every operator on separable Hilbert space can be realized as the compression of the Bergman shift on $M\ominus N$, where $M$ and $N$ are invariant subspaces, $N\subset M$. But a Beurling-type theorem is precisely what the present paper delivers. Given an invariant subspace $M$ in $L^2_a$, consider the subspace $M\ominus TM$, where $T$ stands for multiplication by $z$. This makes sense because $TM$ is a closed subspace of $M$. In Beurling's $H^2$ case, $M\ominus TM$ is one-dimensional and spanned by an inner function. In the $L^2_a$ setting, the dimension of $M\ominus TM$ may be arbitrarily large, even infinite. However, with the correct analogous definition of inner functions in $L^2_a$, all vectors of unit norm in\break $M\ominus TM$ are $L^2_a$-inner. Following Halmos, the subspace $M\ominus TM$ is called the wandering subspace of $M$. Given an invariant subspace, a natural question is: which collections of elements generate it? In particular, one can ask for the least number of elements in a set of generators. It is known that the dimension of the wandering subspace represents a lower bound for the least number of generators. In the paper, it is shown that any orthonormal basis in the wandering subspace (which then consists of $L^2_a$-inner functions) generates $M$ as an invariant subspace. This settles the issue of the minimal number of generators. Let $P$ be the orthogonal projection $M\to M\ominus TM$, and let $L\colon M\to M$ be the operator such that $TL$ is the orthogonal projection $M\to TM$. Then, for $f\in M$, $f=Pf+TLf$. If we do the same for $Lf\in M$, we get $Lf=PLf+TL^2f$ and, inserting it into the original relation for $f$, we get $f=Pf+TPLf+T^2L^2f$. As we go on repeating this process, we get $f=Pf+TPLf+T^2PL^2f+\cdots+T^{n-1}PL^{n-1}f+T^nL^nf$. The point with this decomposition is that, apart from the last term, each term is of the form $T$ to some power times an element of $M\ominus TM$, so that $Pf+TPLf+T^2PL^2f+\cdots+T^{n-1}PL^{n-1}f$ is in\break $[M\ominus TM]$, the invariant subspace generated by $M\ominus TM$. If the operators $T^nL^n$ happened to be uniformly bounded, as they are in the case of $H^2$, $T^nL^nf$ would tend to $0$ in the weak topology, and $f$ would be in the weak closure of $[M\ominus TM]$, which by standard functional analysis coincides with $[M\ominus TM]$. However, for the Bergman space, it seems unlikely that the $T^nL^n$ are uniformly bounded for all possible invariant subspaces $M$, although no immediate counterexample comes to mind. For this reason, the authors try Abel summation instead, and consider for $0
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| 9. |
- Aleman, Alexandru, et al.
(författare)
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Characterizations of a limiting class B∞ of Békollé–Bonami weights
- 2019
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Ingår i: Revista Matematica Iberoamericana. - : European Mathematical Society - EMS - Publishing House GmbH. - 0213-2230. ; 35:6, s. 1677-1692
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Tidskriftsartikel (refereegranskat)abstract
- We explore properties of the class of Békollé–Bonami weights B∞ introduced by the authors in a previous work. Although Békollé–Bonami weights are known to be ill-behaved because they do not satisfy a reverse Hölder property, we prove that when restricting to a class of weights that are “nearly constant on top halves”, one recovers some of the classical properties of Muckenhoupt weights. We also provide an application of this result to the study of the spectra of certain integral operators.
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| 10. |
- Aleman, Alexandru, et al.
(författare)
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Composition of analytic paraproducts
- 2022
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Ingår i: Journal des Mathematiques Pures et Appliquees. - : Elsevier BV. - 0021-7824. ; 158, s. 293-319
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Tidskriftsartikel (refereegranskat)abstract
- For a fixed analytic function g on the unit disc D, we consider the analytic paraproducts induced by g, which are defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=f(z)g(z). The boundedness of these operators on various spaces of analytic functions on D is well understood. The original motivation for this work is to understand the boundedness of compositions of two of these operators, for example Tg2,TgSg,MgTg, etc. Our methods yield a characterization of the boundedness of a large class of operators contained in the algebra generated by these analytic paraproducts acting on the classical weighted Bergman and Hardy spaces in terms of the symbol g. In some cases it turns out that this property is not affected by cancellation, while in others it requires stronger and more subtle restrictions on the oscillation of the symbol g than the case of a single paraproduct.
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