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Artykuły w czasopismach na temat „Analytic Hilbert Spaces”

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Data publikacji: 6 września 2023

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1

de Branges, Louis. "Nodal Hilbert spaces of analytic functions". Journal of Mathematical Analysis and Applications 108, nr 2 (czerwiec 1985): 447–65. http://dx.doi.org/10.1016/0022-247x(85)90038-1.

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2

Wang, Kai. "Analytic Extension of Functions from Analytic Hilbert Spaces*". Chinese Annals of Mathematics, Series B 28, nr 3 (30.04.2007): 321–26. http://dx.doi.org/10.1007/s11401-005-0526-9.

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3

Upmeier, Harald. "Hilbert modules and complex analytic fibre spaces". Rendiconti Lincei - Matematica e Applicazioni 32, nr 3 (16.12.2021): 565–91. http://dx.doi.org/10.4171/rlm/949.

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4

Chen, Xiaoman, Kunyu Guo i Shengzhao Hou. "Analytic Hilbert Spaces over the Complex Plane". Journal of Mathematical Analysis and Applications 268, nr 2 (kwiecień 2002): 684–700. http://dx.doi.org/10.1006/jmaa.2001.7839.

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5

Burtnyak, I., I. Chernega, V. Hladkyi, O. Labachuk i Z. Novosad. "Application of symmetric analytic functions to spectra of linear operators". Carpathian Mathematical Publications 13, nr 3 (11.12.2021): 701–10. http://dx.doi.org/10.15330/cmp.13.3.701-710.

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The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.

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6

Chen, Qiuhui, Luoqing Li i Weibin Wu. "Amplitude spaces of mono-components from Blaschke products and an intrinsic multiresolution analysis". International Journal of Wavelets, Multiresolution and Information Processing 18, nr 06 (15.09.2020): 2050051. http://dx.doi.org/10.1142/s0219691320500514.

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A mono-component is a real-variable and complex-valued analytic signal with nonnegative frequency components. The amplitude of an analytic signal is determined by its phase in a canonical amplitude-phase modulation. This paper investigates the amplitude spaces of analytic signals in terms of the Blaschke products with zeros in [Formula: see text]. It is proved that these amplitude spaces are invariant under the Hilbert transform and form a multiresolution analysis in the Hilbert space of signals with finite energy.

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7

Hou, Shengzhao, i Shuyun Wei. "Ordered analytic Hilbert spaces over the unit disk". Studia Mathematica 185, nr 2 (2008): 127–42. http://dx.doi.org/10.4064/sm185-2-2.

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8

Lanucha, Bartosz, Maria Nowak i Miroslav Pavlovic. "Hilbert matrix operator on spaces of analytic functions". Annales Academiae Scientiarum Fennicae Mathematica 37 (luty 2012): 161–74. http://dx.doi.org/10.5186/aasfm.2012.3715.

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9

Yousefi, B. "Multiplication operators on Hilbert spaces of analytic functions". Archiv der Mathematik 83, nr 6 (grudzień 2004): 536–39. http://dx.doi.org/10.1007/s00013-004-1040-0.

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10

Guo, Kunyu. "Characteristic Spaces and Rigidity for Analytic Hilbert Modules". Journal of Functional Analysis 163, nr 1 (kwiecień 1999): 133–51. http://dx.doi.org/10.1006/jfan.1998.3380.

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11

Mozhyrovska, Zoryana, i Andriy V. Zagorodnyuk. "Hypercyclic Behavior of Translation Operators on Spaces of Analytic Functions on Hilbert Spaces". Journal of Function Spaces 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/139289.

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We consider special Hilbert spaces of analytic functions of many infinite variables and examine composition operators on these spaces. In particular, we prove that under some conditions a translation operator is bounded and hypercyclic.

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12

Pollicott, Mark, i Benedict Sewell. "Explicit examples of resonances for Anosov maps of the torus". Nonlinearity 36, nr 1 (18.11.2022): 110–32. http://dx.doi.org/10.1088/1361-6544/ac9a2e.

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Abstract In (2017 Nonlinearity 30 2667–86) Slipantschuk, Bandtlow and Just gave concrete examples of Anosov diffeomorphisms of T 2 for which their resonances could be completely described. Their approach was based on composition operators acting on analytic anisotropic Hilbert spaces. In this note we present a construction of alternative anisotropic Hilbert spaces which helps to simplify parts of their analysis and gives scope for constructing further examples.

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13

Lopushansky, O. V., i A. V. Zagorodnyuk. "Hilbert spaces of analytic functions of infinitely many variables". Annales Polonici Mathematici 81, nr 2 (2003): 111–22. http://dx.doi.org/10.4064/ap81-2-2.

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14

WATANABE, Shigeru. "Hilbert Spaces of Analytic Functions and the Gegenbauer Polynomials". Tokyo Journal of Mathematics 13, nr 2 (grudzień 1990): 421–27. http://dx.doi.org/10.3836/tjm/1270132271.

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15

Iancu, G. M., i M. W. Wong. "Analytic semigroups and semilinear heat equations in hilbert spaces". Applicable Analysis 69, nr 3-4 (lipiec 1998): 265–83. http://dx.doi.org/10.1080/00036819808840662.

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16

Fricain, Emmanuel, Javad Mashreghi i Daniel Seco. "Cyclicity in Reproducing Kernel Hilbert Spaces of Analytic Functions". Computational Methods and Function Theory 14, nr 4 (4.06.2014): 665–80. http://dx.doi.org/10.1007/s40315-014-0073-z.

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17

Tomin, N. G. "Analytic form of isometric operators in weighted Hilbert spaces". Russian Mathematical Surveys 49, nr 2 (30.04.1994): 182–83. http://dx.doi.org/10.1070/rm1994v049n02abeh002232.

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18

Song, Yisheng, i Liqun Qi. "Infinite-dimensional Hilbert tensors on spaces of analytic functions". Communications in Mathematical Sciences 15, nr 7 (2017): 1897–911. http://dx.doi.org/10.4310/cms.2017.v15.n7.a5.

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19

Domenig, Thomas, Hans Jarchow i Reinhard Riedl. "The domain space of an analytic composition operator". Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 66, nr 1 (luty 1999): 56–65. http://dx.doi.org/10.1017/s1446788700036272.

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AbstractIn this paper we show that, for analytic composition operators between weighted Bergman spaces (including Hardy spaces) and as far as boundedness, compactness, order boundedness and certain summing properties of the adjoint are concerned, it is possible to modify domain spaces in a systematic fashion: there is a space of analytic functions which embeds continuously into each of the spaces under consideration and on which the above properties of the operator are decided.A remarkable consequence is that, in the setting of composition operators between weighted Bergman spaces, the properties in question can be identified as properties of the operator as a map between appropriately chosen Hilbert spaces.

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20

Tong, Cezhong, Zhan Zhang i Biao Xu. "Simply connected topological spaces of weighted composition operators". Open Mathematics 18, nr 1 (2.12.2020): 1440–50. http://dx.doi.org/10.1515/math-2020-0082.

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21

Al-Rawashdeh, Waleed. "Generalized Composition Operators on Weighted Hilbert Spaces of Analytic Functions". International Journal of Advanced Research in Mathematics 10 (wrzesień 2017): 1–13. http://dx.doi.org/10.18052/www.scipress.com/ijarm.10.1.

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Letφbe an analytic self-map of the open unit disk D andgbe an analytic function on D. The generalized composition operator induced by the mapsgandφis defined by the integral operatorI(g,φ)f(z) =∫0zf′(φ(ς))g(ς)dς. Given an admissible weightω, the weighted Hilbert spaceHωconsists of all analytic functionsfsuch that ∥f∥2Hω= |f(0)|2+∫D|f′(z)|2ω(z)dA(z) is finite. In this paper, we characterize the boundedness and compactness of the generalized composition operators on the spaceHωusing theω-Carleson measures. Moreover, we give a lower bound for the essential norm of these operators.

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22

Fricain, Emmanuel, Javad Mashreghi i Rishika Rupam. "Backward shift invariant subspaces in reproducing kernel Hilbert spaces". MATHEMATICA SCANDINAVICA 126, nr 1 (29.03.2020): 142–60. http://dx.doi.org/10.7146/math.scand.a-119120.

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In this note, we describe the backward shift invariant subspaces for an abstract class of reproducing kernel Hilbert spaces. Our main result is inspired by a result of Sarason concerning de Branges-Rovnyak spaces (the non-extreme case). Furthermore, we give new applications in the context of the range space of co-analytic Toeplitz operators and sub-Bergman spaces.

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23

Mosaleheh, K., i K. Seddighi. "Interpolation by multipliers on certain spaces of analytic functions". International Journal of Mathematics and Mathematical Sciences 23, nr 8 (2000): 547–54. http://dx.doi.org/10.1155/s0161171200000958.

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Letℋbe a Hilbert space of analytic functions on a planar domainGsuch that, for eachλinG, the linear functionaleλof evaluation atλis bounded onℋ. Furthermore, assume thatzℋ⊂ℋandσ(Mz)=G¯is anM-spectral set forMz, the operator of multiplication byz. This paper is devoted to the study of interpolation by multipliers of the spaceℋand, in particular, the Dirichlet space.

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24

Vlasenko, Larisa A., Anatoly G. Rutkas i Arkady A. Chikrii. "STOCHASTIC DIFFERENTIAL GAMES IN DISTRIBUTED SYSTEMS WITH DELAY". Journal of Automation and Information sciences 1 (1.01.2021): 41–54. http://dx.doi.org/10.34229/0572-2691-2021-1-4.

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We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.

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25

Aleman, Alexandru, i Bartosz Malman. "Hilbert spaces of analytic functions with a contractive backward shift". Journal of Functional Analysis 277, nr 1 (lipiec 2019): 157–99. http://dx.doi.org/10.1016/j.jfa.2018.08.019.

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26

Aleman, A. "Finite Codimensional Invariant Subspaces in Hilbert Spaces of Analytic Functions". Journal of Functional Analysis 119, nr 1 (styczeń 1994): 1–18. http://dx.doi.org/10.1006/jfan.1994.1001.

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27

Martín, María J., i Dragan Vukotić. "Adjoints of composition operators on Hilbert spaces of analytic functions". Journal of Functional Analysis 238, nr 1 (wrzesień 2006): 298–312. http://dx.doi.org/10.1016/j.jfa.2006.04.024.

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28

Carlsson, Marcus. "Boundary behavior in Hilbert spaces of vector-valued analytic functions". Journal of Functional Analysis 247, nr 1 (czerwiec 2007): 169–201. http://dx.doi.org/10.1016/j.jfa.2007.02.006.

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29

Cheng, Guozheng, Kunyu Guo i Kai Wang. "Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces". Journal of Functional Analysis 258, nr 12 (czerwiec 2010): 4229–50. http://dx.doi.org/10.1016/j.jfa.2010.01.021.

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30

Wei, Shuyun. "Beurling’s phenomenon on analytic Hilbert spaces over the complex plane". Proceedings of the American Mathematical Society 138, nr 04 (1.04.2010): 1439. http://dx.doi.org/10.1090/s0002-9939-09-10196-x.

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31

Alpay, Daniel, i Harry Dym. "Hilbert spaces of analytic functions, inverse scattering and operator models.II". Integral Equations and Operator Theory 8, nr 2 (marzec 1985): 145–80. http://dx.doi.org/10.1007/bf01202812.

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32

Kellay, Karim, i Pascal Lefèvre. "Compact composition operators on weighted Hilbert spaces of analytic functions". Journal of Mathematical Analysis and Applications 386, nr 2 (luty 2012): 718–27. http://dx.doi.org/10.1016/j.jmaa.2011.08.033.

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33

Solimini, Sergio, i Cyril Tintarev. "Concentration analysis in Banach spaces". Communications in Contemporary Mathematics 18, nr 03 (22.03.2016): 1550038. http://dx.doi.org/10.1142/s0219199715500388.

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The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of [Formula: see text]-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and [Formula: see text]-spaces, but not in [Formula: see text], [Formula: see text]. [Formula: see text]-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of [Formula: see text]-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.

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34

Wang, Zhi-jie, Jie-sheng Xiao i Xian-min Xu. "Beurling-Type Theorem for a Class of Reproducing Kernel Hilbert Spaces". Journal of Mathematics 2022 (29.01.2022): 1–5. http://dx.doi.org/10.1155/2022/4478452.

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35

Berg, Michael C. "Derived Categories and the Analytic Approach to General Reciprocity Laws: Part III". International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–19. http://dx.doi.org/10.1155/2010/731093.

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Building on the scaffolding constructed in the first two articles in this series, we now proceed to the geometric phase of our sheaf (-complex) theoretic quasidualization of Kubota's formalism forn-Hilbert reciprocity. Employing recent work by Bridgeland on stability conditions, we extend our yoga oft-structures situated above diagrams of specifically designed derived categories to arrangements of metric spaces or complex manifolds. This prepares the way for provingn-Hilbert reciprocity by means of singularity analysis.

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36

Mirzakarimi, G., i K. Seddighi. "Weighted Composition Operators on Bergman and Dirichlet Spaces". gmj 4, nr 4 (sierpień 1997): 373–83. http://dx.doi.org/10.1515/gmj.1997.373.

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Abstract Let 𝐻(Ω) denote a functional Hilbert space of analytic functions on a domain Ω. Let 𝑤 : Ω → 𝐂 and ϕ : Ω → Ω be such that 𝑤 𝑓 ○ ϕ is in 𝐻(Ω) for every 𝑓 in 𝐻(Ω). The operator 𝑤𝐶 ϕ Given by 𝑓 → 𝑤 𝑓 ○ ϕ is called a weighted composition operator on 𝐻(Ω). In this paper we characterize such operators and those for which (𝑤𝐶 ϕ )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.

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37

Sadraoui, Houcine. "Hyponormality on general Bergman spaces". Filomat 33, nr 17 (2019): 5737–41. http://dx.doi.org/10.2298/fil1917737s.

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A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

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38

Xu, Yun, i Shanli Ye. "A Derivative Hilbert operator acting from Bergman spaces to Hardy spaces". AIMS Mathematics 8, nr 4 (2023): 9290–302. http://dx.doi.org/10.3934/math.2023466.

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<abstract><p>Let $ \mu $ be a positive Borel measure on the interval $ [0, 1) $. The Hankel matrix $ \mathcal{H}_{\mu} = (\mu_{n, k})_{n, k\geq 0} $ with entries $ \mu_{n, k} = \mu_{n+k} $, where $ \mu_{n} = \int_{[0, 1)}t^nd\mu(t) $, formally induces the operator as follows:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal{DH}_\mu(f)(z) = \sum\limits_{n = 0}^\infty\left(\sum\limits_{k = 0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , \; z\in \mathbb{D}, $\end{document} </tex-math></disp-formula></p> <p>where $ f(z) = \sum_{n = 0}^\infty a_nz^n $ is an analytic function in $ \mathbb{D} $. In this article, we characterize those positive Borel measures on $ [0, 1) $ such that $ \mathcal{DH}_\mu $ is bounded (resp., compact) from Bergman spaces $ \mathcal{A}^p $ into Hardy spaces $ H^q $, where $ 0 < p, q < \infty $.</p></abstract>

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39

Fricain, Emmanuel, Andreas Hartmann i William T. Ross. "Range Spaces of Co-Analytic Toeplitz Operators". Canadian Journal of Mathematics 70, nr 6 (20.11.2018): 1261–83. http://dx.doi.org/10.4153/cjm-2017-057-4.

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AbstractIn this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

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40

Yousefi, Bahmann, i Javad Izadi. "Weighted Composition Operators and Supercyclicity Criterion". International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–5. http://dx.doi.org/10.1155/2011/514370.

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We consider an equivalent condition to the property of Supercyclicity Criterion, and then we investigate this property for the adjoint of weighted composition operators acting on Hilbert spaces of analytic functions.

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41

Seddighi, K., i S. M. Vaezpour. "Commutants of certain multiplication operators on Hilbert spaces of analytic functions". Studia Mathematica 133, nr 2 (1999): 121–30. http://dx.doi.org/10.4064/sm-133-2-121-130.

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42

Chen, Yong. "Quasi-wandering subspaces in a class of reproducing analytic Hilbert spaces". Proceedings of the American Mathematical Society 140, nr 12 (1.12.2012): 4235–42. http://dx.doi.org/10.1090/s0002-9939-2012-11290-0.

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43

Sun, Hong-Wei, i Ding-Xuan Zhou. "Reproducing Kernel Hilbert Spaces Associated with Analytic Translation-Invariant Mercer Kernels". Journal of Fourier Analysis and Applications 14, nr 1 (24.01.2008): 89–101. http://dx.doi.org/10.1007/s00041-007-9003-z.

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44

Stochel, Jan, i Jerzy Bartłomiej Stochel. "Composition operators on Hilbert spaces of entire functions with analytic symbols". Journal of Mathematical Analysis and Applications 454, nr 2 (październik 2017): 1019–66. http://dx.doi.org/10.1016/j.jmaa.2017.05.021.

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45

Gröchenig, Karlheinz, i Joaquim Ortega-Cerdà. "Marcinkiewicz–Zygmund Inequalities for Polynomials in Bergman and Hardy Spaces". Journal of Geometric Analysis 31, nr 7 (3.02.2021): 7595–619. http://dx.doi.org/10.1007/s12220-020-00599-5.

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AbstractWe study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz–Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly different behavior.

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46

Galindo, P., T. W. Gamelin i Mikael Lindström. "Spectra of composition operators on algebras of analytic functions on Banach spaces". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, nr 1 (luty 2009): 107–21. http://dx.doi.org/10.1017/s0308210507000819.

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Let E be a Banach space, with unit ball BE. We study the spectrum and the essential spectrum of a composition operator on H∞(BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

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47

Cheng, Guozheng, i Xiang Fang. "An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions". Journal of Functional Analysis 260, nr 7 (kwiecień 2011): 2027–42. http://dx.doi.org/10.1016/j.jfa.2010.09.015.

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48

Bekbayev, N. T., i K. S. Tulenov. "On boundedness of the Hilbert transform on Marcinkiewicz spaces". BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 100, nr 4 (30.12.2020): 26–32. http://dx.doi.org/10.31489/2020m4/26-32.

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We study boundedness properties of the classical (singular) Hilbert transform (Hf)(t) = p.v.1/π \int_R f(s)/(t − s)ds acting on Marcinkiewicz spaces. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Questions involving the H arise therefore from the utilization of complex methods in Fourier analysis, for example. In particular, the H plays the crucial role in questions of norm-convergence of Fourier series and Fourier integrals. We consider the problem of what is the least rearrangement-invariant Banach function space F(R) such that H : Mφ(R) → F(R) is bounded for a fixed Marcinkiewicz space Mφ(R). We also show the existence of optimal rearrangement-invariant Banach function range on Marcinkiewicz spaces. We shall be referring to the space F(R) as the optimal range space for the operator H restricted to the domain Mφ(R) ⊆ Λϕ0(R). Similar constructions have been studied by J.Soria and P.Tradacete for the Hardy and Hardy type operators [1]. We use their ideas to obtain analogues of their some results for the H on Marcinkiewicz spaces.

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49

Lo, Ching-on, i Anthony Wai-keung Loh. "Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces". Opuscula Mathematica 40, nr 4 (2020): 495–507. http://dx.doi.org/10.7494/opmath.2020.40.4.495.

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Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).

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50

Blasco, Oscar. "A characterization of Hilbert spaces in terms of multipliers between spaces of vector-valued analytic functions." Michigan Mathematical Journal 42, nr 3 (1995): 537–43. http://dx.doi.org/10.1307/mmj/1029005311.

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