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Journal articles on the topic 'Hilbert space operators'

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Published: 6 September 2023

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1

Agniel, Vidal. "Unitary skew-dilations of Hilbert space operators." Extracta Mathematicae 35, no. 2 (2020): 137–84. http://dx.doi.org/10.17398/2605-5686.35.2.137.

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2

Kérchy, László. "Pluquasisimilar Hilbert space operators." Acta Scientiarum Mathematicarum 86, no. 34 (2020): 503–20. http://dx.doi.org/10.14232/actasm-020-973-4.

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3

Krnić, Mario. "Hilbert-type inequalities for Hilbert space operators." Quaestiones Mathematicae 36, no. 2 (June 2013): 209–23. http://dx.doi.org/10.2989/16073606.2013.801148.

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4

Jarchow, Hans. "Factoring absolutely summing operators through Hilbert-Schmidt operators." Glasgow Mathematical Journal 31, no. 2 (May 1989): 131–35. http://dx.doi.org/10.1017/s0017089500007643.

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Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.
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5

Carmo, Joao R., and S. Waleed Noor. "Universal composition operators." Journal of Operator Theory 87, no. 1 (December 15, 2021): 137–56. http://dx.doi.org/10.7900/jot.2020aug03.2301.

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A Hilbert space operator U is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0
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6

Rugiri, Peter Githara. "Spectrum of bounded operators in Hilbert spaces." Editon Consortium Journal of Physical and Applied Sciences 3, no. 1 (August 31, 2023): 102–8. http://dx.doi.org/10.51317/ecjpas.v3i1.411.

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This paper sought to study the spectrum operators by emphasising on condition of commuting operators so as to expose classes of operators. Here study of various classes of bounded operators on a Hilbert space H is one of the most important topics in the preparation of the study of the Hilbert spaces. In case a abounded operator A commutes at least with its own adjoint A* it forms important classes of operators on H, eg normal, unitary, self – adjoint etc. The operators under the study are bounded operators operating in a complete space called Hilbert spaces.
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7

Vaccaro, John A. "Phase operators on Hilbert space." Physical Review A 51, no. 4 (April 1, 1995): 3309–17. http://dx.doi.org/10.1103/physreva.51.3309.

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8

Leung, Denny H. "Factoring operators through Hilbert space." Israel Journal of Mathematics 71, no. 2 (December 1990): 225–27. http://dx.doi.org/10.1007/bf02811886.

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9

Boenkost, W., and F. Constantinescu. "Vertex operators in Hilbert space." Journal of Mathematical Physics 34, no. 8 (August 1993): 3607–15. http://dx.doi.org/10.1063/1.530048.

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10

Khrushchev, Sergei, and Vladimir Peller. "Hankel operators on Hilbert space." Acta Applicandae Mathematicae 5, no. 1 (January 1986): 96–100. http://dx.doi.org/10.1007/bf00049173.

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11

MacLennan, Bruce James. "Cognition in Hilbert space." Behavioral and Brain Sciences 36, no. 3 (May 14, 2013): 296–97. http://dx.doi.org/10.1017/s0140525x1200283x.

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AbstractUse of quantum probability as a top-down model of cognition will be enhanced by consideration of the underlying complex-valued wave function, which allows a better account of interference effects and of the structure of learned and ad hoc question operators. Furthermore, the treatment of incompatible questions can be made more quantitative by analyzing them as non-commutative operators.
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12

JUNG, IL BONG, EUNGIL KO, and CARL PEARCY. "ALMOST INVARIANT HALF-SPACES FOR OPERATORS ON HILBERT SPACE." Bulletin of the Australian Mathematical Society 97, no. 1 (August 31, 2017): 133–40. http://dx.doi.org/10.1017/s0004972717000533.

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The theory of almost invariant half-spaces for operators on Banach spaces was begun recently and is now under active development. Much less attention has been given to almost invariant half-spaces for operators on Hilbert space, where some techniques and results are available that are not present in the more general context of Banach spaces. In this note, we begin such a study. Our much simpler and shorter proofs of the main theorems have important consequences for the matricial structure of arbitrary operators on Hilbert space.
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13

kian, Mohsen. "Hardy--Hilbert type inequalities for Hilbert space operators." Annals of Functional Analysis 3, no. 2 (2012): 128–34. http://dx.doi.org/10.15352/afa/1399899937.

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14

Moghaddam, Sadaf Fakri, and Alireza Kamel Mirmostafaee. "Numerical Radius of Bounded Operators with ℓ p -Norm." Tatra Mountains Mathematical Publications 81, no. 1 (November 1, 2022): 155–64. http://dx.doi.org/10.2478/tmmp-2022-0012.

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Abstract We study the numerical radius of bounded operators on direct sum of a family of Hilbert spaces with respect to the ℓ p -norm, where 1 ≤ p ≤∞. We propose a new method which enables us to prove validity of many inequalities on numerical radius of bounded operators on Hilbert spaces when the underling space is a direct sum of Hilbert spaces with ℓ p -norm, where 1 ≤ p ≤ 2. We also provide an example to show that some known results on numerical radius are not true for a space that is the set of bounded operators on ℓ p -sum of Hilbert spaces where 2 <p < ∞. We also present some applications of our results.
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15

Bhardwaj, Ruchi, S. K. Sharma, and S. K. Kaushik. "Trace class operators via OPV-frames." Filomat 35, no. 13 (2021): 4353–68. http://dx.doi.org/10.2298/fil2113353b.

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Trace class operators for quaternionic Hilbert spaces (QHS) were studied by Moretti and Oppio [18]. In this paper, we study trace class operators via operator valued frames (OPV-frames). We introduce OPV-frames in a right quaternionic Hilbert space H with range in a two sided quaternionic Hilbert space K and obtain various results including several characterizations of OPV-frames. Also, we obtain a necessary and sufficient condition for a bounded operator on a right QHS to be a trace class operator which generalizes a similar result by Attal [2]. Moreover, we construct a trace class operator on a two sided QHS. Finally, we study quaternionic quantum channels as completely positive trace preserving maps and obtain various Choi-Kraus type representations of quaternionic quantum channels using OPV-frames in quaternionic Hilbert spaces.
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16

Alshammari, Hadi Obaid. "Extension of m-Symmetric Hilbert Space Operators." Journal of Mathematics 2022 (December 28, 2022): 1–8. http://dx.doi.org/10.1155/2022/5272632.

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We introduce a new class of operators, which we will call the class of P -quasi- m -symmetric operators that includes m -symmetric operators and k -quasi m -symmetric operators. Some basic structural properties of this class of operators are established based on the operator matrix representation associated with such operators.
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17

Burtnyak, I., I. Chernega, V. Hladkyi, O. Labachuk, and Z. Novosad. "Application of symmetric analytic functions to spectra of linear operators." Carpathian Mathematical Publications 13, no. 3 (December 11, 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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18

Roopaei, Hadi. "Bounds of operators on the Hilbert sequence space." Concrete Operators 7, no. 1 (September 3, 2020): 155–65. http://dx.doi.org/10.1515/conop-2020-0104.

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AbstractThe author has computed the bounds of the Hilbert operator on some sequence spaces [18, 19]. Through this study the author has investigated the bounds of operators on the Hilbert sequence space and the present study is a complement of those previous research.
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19

Cabrera, M., J. Martínez, and A. Rodríguez. "Hilbert modules revisited: orthonormal bases and Hilbert-Schmidt operators." Glasgow Mathematical Journal 37, no. 1 (January 1995): 45–54. http://dx.doi.org/10.1017/s0017089500030378.

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The concept of a Hilbert module (over an H*-algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H*-algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.
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20

Novosad, Z. H. "Topological transitivity of translation operators in a non-separable Hilbert space." Carpathian Mathematical Publications 15, no. 1 (June 30, 2023): 278–85. http://dx.doi.org/10.15330/cmp.15.1.278-285.

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We consider a Hilbert space of entire analytic functions on a non-separable Hilbert space, associated with a non-separable Fock space. We show that under some conditions operators, like the differentiation operators and translation operators, are topologically transitive in this space.
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21

Babenko, V. F., N. V. Parfinovych, and D. S. Skorokhodov. "The best approximation of closed operators by bounded operators in Hilbert spaces." Carpathian Mathematical Publications 14, no. 2 (December 30, 2022): 453–63. http://dx.doi.org/10.15330/cmp.14.2.453-463.

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We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.
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22

Abderramán Marrero, J., and V. Tomeo. "Arrowhead operators on a Hilbert space." Operators and Matrices, no. 3 (2016): 593–609. http://dx.doi.org/10.7153/oam-10-34.

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23

Kérchy, László. "Unitary asymptotes of Hilbert space operators." Banach Center Publications 30, no. 1 (1994): 191–201. http://dx.doi.org/10.4064/-30-1-191-201.

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24

Cegielski, Andrzej, and Yair Censor. "On Componental Operators in Hilbert Space." Numerical Functional Analysis and Optimization 42, no. 13 (October 3, 2021): 1555–71. http://dx.doi.org/10.1080/01630563.2021.2006695.

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25

Yakubovich, Dmitry, and Sameer Chavan. "Spherical tuples of Hilbert space operators." Indiana University Mathematics Journal 64, no. 2 (2015): 577–612. http://dx.doi.org/10.1512/iumj.2015.64.5471.

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26

Morassaei, A., F. Mirzapour, and M. S. Moslehian. "Bellman inequality for Hilbert space operators." Linear Algebra and its Applications 438, no. 10 (May 2013): 3776–80. http://dx.doi.org/10.1016/j.laa.2011.06.042.

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27

Hirzallah, Omar. "Commutator inequalities for Hilbert space operators." Linear Algebra and its Applications 431, no. 9 (October 2009): 1571–78. http://dx.doi.org/10.1016/j.laa.2009.05.026.

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28

Khatskevich, V. A., M. I. Ostrovskii, and V. S. Shulman. "Quadratic Inequalities for Hilbert Space Operators." Integral Equations and Operator Theory 59, no. 1 (June 27, 2007): 19–34. http://dx.doi.org/10.1007/s00020-007-1511-3.

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29

Prǎjiturǎ, Gabriel T. "Irregular vectors of Hilbert space operators." Journal of Mathematical Analysis and Applications 354, no. 2 (June 2009): 689–97. http://dx.doi.org/10.1016/j.jmaa.2009.01.034.

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30

Glasser, M. L. "Exponentials of Certain Hilbert Space Operators." SIAM Review 33, no. 3 (September 1991): 472. http://dx.doi.org/10.1137/1033103.

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31

ć Hot, Jadranka Mić, Josip Pečarić, and Marjan Praljak. "Levinson's inequality for Hilbert space operators." Journal of Mathematical Inequalities, no. 4 (2015): 1271–85. http://dx.doi.org/10.7153/jmi-09-97.

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32

Drivaliaris, Dimosthenis, and Nikos Yannakakis. "Hilbert space structure and positive operators." Journal of Mathematical Analysis and Applications 305, no. 2 (May 2005): 560–65. http://dx.doi.org/10.1016/j.jmaa.2004.12.007.

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33

Cheung, Wing-Sum, and Josip Pečarić. "Bohr's inequalities for Hilbert space operators." Journal of Mathematical Analysis and Applications 323, no. 1 (November 2006): 403–12. http://dx.doi.org/10.1016/j.jmaa.2005.10.046.

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34

Sababheh, Mohammad, and Hamid Reza Moradi. "New orders among Hilbert space operators." Mathematical Inequalities & Applications, no. 2 (2023): 415–32. http://dx.doi.org/10.7153/mia-2023-26-27.

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35

Amson, J. C., and N. Gopal Reddy. "A Hilbert algebra of Hilbert-Schmidt quadratic operators." Bulletin of the Australian Mathematical Society 41, no. 1 (February 1990): 123–34. http://dx.doi.org/10.1017/s0004972700017913.

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A quadratic operator Q of Hilbert-Schmidt class on a real separable Hilbert space H is shown to be uniquely representable as a sequence of self-adjoint linear operators of Hilbert-Schmidt class on H, such that Q(x) = Σk〈Lkx, x〉uk with respect to a Hilbert basis . It is shown that with the norm | ‖Q‖ | = (Σk ‖Lk‖2)½ and inner-product 〈〈〈Q, P〉〉〉 = Σk 〈〈Lk, Mk〉〉, together with a multiplication defined componentwise through the composition of the linear components, the vector space of all Hilbert-Schmidt quadratic operators Q on H becomes a linear H*-algebra containing an ideal of nuclear (trace class) quadratic operators. In the finite dimensional case, each Q is also shown to have another representation as a block-diagonal matrix of Hilbert-Schmidt class which simplifies the practical computation and manipulation of quadratic operators.
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36

Holub, J. R. "On Shift Operators." Canadian Mathematical Bulletin 31, no. 1 (March 1, 1988): 85–94. http://dx.doi.org/10.4153/cmb-1988-013-8.

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AbstractA definition of an isometric shift operator on a Banach space is given which extends the usual definition of a shift operator on a separable Hilbert space. It is shown that there is no such shift on many of the common Banach spaces of continuous functions. The associated ideas of a semi-shift and a backward shift are also introduced and studied in the case of continuous function spaces.
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37

Minculete, Nicuşor. "About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators." Symmetry 13, no. 2 (February 11, 2021): 305. http://dx.doi.org/10.3390/sym13020305.

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The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.
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38

Ploymukda, Arnon, and Pattrawut Chansangiam. "Norm estimations, continuity, and compactness for Khatri-Rao products of Hilbert Space operators." Malaysian Journal of Fundamental and Applied Sciences 14, no. 4 (December 16, 2018): 382–86. http://dx.doi.org/10.11113/mjfas.v14n4.881.

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We provide estimations for the operator norm, the trace norm, and the Hilbert-Schmidt norm for Khatri-Rao products of Hilbert space operators. It follows that the Khatri-Rao product is continuous on norm ideals of compact operators equipped with the topologies induced by such norms. Moreover, if two operators are represented by block matrices in which each block is nonzero, then their Khatri-Rao product is compact if and only if both operators are compact. The Khatri-Rao product of two operators are trace-class (Hilbert-Schmidt class) if and only if each factor is trace-class (Hilbert-Schmidt class, respectively).
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39

TRAPANI, C. "QUASI *-ALGEBRAS OF OPERATORS AND THEIR APPLICATIONS." Reviews in Mathematical Physics 07, no. 08 (November 1995): 1303–32. http://dx.doi.org/10.1142/s0129055x95000475.

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The main facts of the theory of quasi*-algebras of operators acting in a rigged Hilbert space are reviewed. The particular case where the rigged Hilbert space is generated by a self-adjoint operator in Hilbert space is examined in more details. A series of applications to quantum theories are discussed.
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40

KällstrÖm, A., and B. D. Sleeman. "Joint spectra for commuting operators." Proceedings of the Edinburgh Mathematical Society 28, no. 2 (June 1985): 233–48. http://dx.doi.org/10.1017/s0013091500022677.

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The theory of joint spectra for commuting operators in a Hilbert space has recently been studied by several authors (Vasilescu [11,12], Curto [4,5], and Cho-Takaguchi[2,3]). In this paper we willuse the definition by Taylor [10] of the joint spectrum to show that thejoint spectrum is determined by the action of certain "Laplacians"(cf. Curto [4,5]) of a chain-complex of Hilbert spaces.
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41

Bercovici, Hari. "Three Test Problems for Quasisimilarity." Canadian Journal of Mathematics 39, no. 4 (August 1, 1987): 880–92. http://dx.doi.org/10.4153/cjm-1987-043-x.

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Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.
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42

Bunao, Joseph, and Eric A. Galapon. "The Bender-Dunne basis operators as Hilbert space operators." Journal of Mathematical Physics 55, no. 2 (February 2014): 022102. http://dx.doi.org/10.1063/1.4863901.

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43

Celeghini, Enrico, Manuel Gadella, and Mariano A. del Olmo. "Groups, Special Functions and Rigged Hilbert Spaces." Axioms 8, no. 3 (July 27, 2019): 89. http://dx.doi.org/10.3390/axioms8030089.

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We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O ( 2 ) and functions on the unit circle, S U ( 2 ) and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, S O ( 3 , 2 ) and spherical harmonics, s u ( 1 , 1 ) and Laguerre functions, s u ( 2 , 2 ) and algebraic Jacobi functions and, finally, s u ( 1 , 1 ) ⊕ s u ( 1 , 1 ) and Zernike functions on a circle.
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44

Wong, M. W. "Minimal and Maximal Operator Theory With Applications." Canadian Journal of Mathematics 43, no. 3 (June 1, 1991): 617–27. http://dx.doi.org/10.4153/cjm-1991-036-7.

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AbstractLetXbe a complex Banach space andAa linear operator fromXintoXwith dense domain. We construct the minimal and maximal operators of the operatorAand prove that they are equal under reasonable hypotheses on the spaceXand operatorA. As an application, we obtain the existence and regularity of weak solutions of linear equations on the spaceX. As another application we obtain a criterion for a symmetric operator on a complex Hilbert space to be essentially self-adjoint. An application to pseudo-differential operators of the Weyl type is given.
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45

McDonald, G., and C. Sundberg. "On the Spectra of Unbounded Subnormal Operators." Canadian Journal of Mathematics 38, no. 5 (October 1, 1986): 1135–48. http://dx.doi.org/10.4153/cjm-1986-057-x.

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Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),
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46

Altwaijry, Najla, Kais Feki, and Shigeru Furuichi. "Generalized Cauchy–Schwarz Inequalities and A-Numerical Radius Applications." Axioms 12, no. 7 (July 22, 2023): 712. http://dx.doi.org/10.3390/axioms12070712.

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The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes a positive semidefinite operator in a complex Hilbert space. Some of our novel A-numerical radius inequalities expand upon the existing literature on numerical radius inequalities with Hilbert space operators, which are important tools in functional analysis. We use techniques from semi-Hilbert space theory to prove our results and highlight some applications of our findings.
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47

Sid Ahmed, Ould Ahmed Mahmoud. "m-ISOMETRIC OPERATORS ON BANACH SPACES." Asian-European Journal of Mathematics 03, no. 01 (March 2010): 1–19. http://dx.doi.org/10.1142/s1793557110000027.

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We introduce the class of m-isometric operators on Banach spaces. This generalizes to Banach space the m-isometric operators on Hilbert space introduced by Agler and Stankus. We establish some basic properties and we introduce the notion of m-invertibility as a natural generalization of the invertibility on Banach spaces.
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48

Matache, Valentin. "Weighted composition operators on Hardy–Smirnov spaces." Concrete Operators 9, no. 1 (January 1, 2022): 160–76. http://dx.doi.org/10.1515/conop-2022-0136.

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Abstract Operators of type f → ψf ◦ φ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on Hardy–Smirnov spaces and prove that their unitarily invariant properties are reducible to the study of weighted composition operators on the classical Hardy space over a disc. We give examples of such results, for instance proving that Forelli’s theorem saying that the isometries of non–Hilbert Hardy spaces over the unit disc need to be special weighted composition operators extends to all non–Hilbert Hardy–Smirnov spaces. A thorough study of boundedness of weighted composition operators is performed.
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49

Al-Muttalibi, Rana, and Radhi M.A Ali. "Certain types of linear operators on probabilistic Hilbert space." Global Journal of Mathematical Analysis 3, no. 2 (May 24, 2015): 81. http://dx.doi.org/10.14419/gjma.v3i2.4664.

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<p>The purpose of this paper is to introduce some definitions, properties and basic results that show the relation between F-bounded of linear operator in probabilistic Hilbert space and bounded operator in norm. In the paper, we prove that the adjoint operator in probabilistic Hilbert space is bounded. The notion of the continuous operators in probabilistic Hilbert space and some basic results are given. In addition, we note that every operator in probabilistic real Hilbert space is a self-adjoint Operator.</p>
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50

Ram, Sonu, and Preeti Dharmarha. "Totally (m, n)-paranormal operators." Gulf Journal of Mathematics 15, no. 1 (August 18, 2023): 57–66. http://dx.doi.org/10.56947/gjom.v15i1.1248.

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In this paper, we investigate the class of totally (m, n)-paranormal operators on Hilbert space and prove some additional properties of totally (m, n)-paranormal operators on Hilbert space. Also, we show that Weyl's theorem holds for operators in this class. Moreover, we also show that for totally (m, n)-paranormal operators, the Weyl spectrum satisfies the spectral mapping theorem.
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