Analytic and entire vectors for representations of Lie groups

Goodman, Roe

doi:10.1090/s0002-9947-1969-0248285-6

Cited by 132 publications

(45 citation statements)

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“…For a number γ ∈ [0, ∞), we put where is a Banach space with respect to the norm \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {N}_{0} = \mathbb {N}\bigcup \lbrace 0\rbrace$\end{document}. The linear space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {G}_{\lbrace \gamma \rbrace }(A) \ (\mathfrak {G}_{(\gamma )}(A))$\end{document} is endowed with the inductive (projective) limit topology of the Banach spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {G}_{\gamma }^{\alpha }(A)$\end{document}: Specifically, the spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {G}_{\lbrace 1\rbrace }(A), \ \mathfrak {G}_{(1)}(A)$\end{document}, and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {G}_{\lbrace 0\rbrace }(A)$\end{document} are not other than the spaces of analytic 18, entire 6 and entire of exponential type 22 vectors of the operator A , respectively (see also 3, 23).…”

Section: Some Subspaces Of Infinitely Differentiable Vectors Of a Clomentioning

confidence: 99%

On extensions and restrictions of semigroups of linear operators in a Banach space and their applications

Горбачук

2012

Mathematische Nachrichten

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Key words Differential equation in a Banach space, classical solution, weak solution, locally convex space, C0 -semigroup, extension and restriction of a C0 -semigroup, exponential operator-valued function, locally analytic vector-valued function, Cauchy problem, Cauchy-Kovalevskaya theorem MSC (2010) 34G10, 47D06 Dedicated to Eduard Tsekanovskiȋ in honour of his 75th birthdayThe description of all solutions on (0, ∞) of differential equations of parabolic and elliptic type in a Banach space is given without imposing any conditions on their behavior near the point 0, and the properties of them are studied. The necessary and sufficient conditions on the initial data, under which the Cauchy problem for a general operator-differential equation is solvable in the class of locally analytic, entire or entire of finite order and finite type vector-valued functions, are established. The results are based on the theory of restrictions and extensions of analytic semigroups set forth below.y Br (B) = max z ∈Or y(z) .

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Section: Some Subspaces Of Infinitely Differentiable Vectors Of a Clomentioning

confidence: 99%

On extensions and restrictions of semigroups of linear operators in a Banach space and their applications

Горбачук

2012

Mathematische Nachrichten

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“…Pour pouvoir appliquer le théorème (1.2), il nous faut démontrer pour les crochets d'ordre j (j^r) une égalité du type (1. On renvoie à (7] pour les définitions. On peut démontrer exactement comme le théorème 1.3 et compte tenu d'un théorème de R. Goodman [7] le théorème suivant : Il est montré dans [2] que le théorème (1.2) est encore vrai dans les classes de Gevrey G 5 .…”

Section: Démonstration Du Théorème 13unclassified

“…On renvoie à (7] pour les définitions. On peut démontrer exactement comme le théorème 1.3 et compte tenu d'un théorème de R. Goodman [7] le théorème suivant : Il est montré dans [2] que le théorème (1.2) est encore vrai dans les classes de Gevrey G 5 . Le théorème suivant se démontre comme le théorème (1.3) : THÉORÈME 2.9.…”

Section: Démonstration Du Théorème 13unclassified

“…telle que, pour tout compact K de Q, il existe une constante CK>O telle que l'on ait, pour i= 1, ..., n et tout k ; sont les champs ô^., le théorème est dû à F. Browder[3]. Dans le cadre des algèbres de Lie, un théorème du même type que le théorème (1.2) est démontré par R. Goodman[7], pour caractériser les vecteurs analytiques d'une représentation unitaire. Nous reviendrons sur cette question dans la remarque (2.6).…”

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Analyticité et itères d'un système de champs non elliptique

Damlakhi¹,

Helffer²

1980

Ann. Sci. École Norm. Sup.

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“…(B), C (n!) (B), and C {1} (B) of analytic vectors[7], entire vectors[8], and entire vectors of exponential type[9] of the operator B.In particular, if H = L 2 (a, b), −∞ < a < b < ∞, B = d/dx, and D(B) is the Sobolev space W 1 2 (a, b), then C ∞ (B)coincides with the set of infinitely differentiable functions on [a, b], C {n!} (B) and C (n!)…”

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On the approximation to solutions of operator equations by the least squares method

Горбачук¹

2005

Funct Anal Its Appl

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We consider the equation Au = f , where A is a linear operator with compact inverse A −1 in a separable Hilbert space H. For the approximate solution u n of this equation by the least squares method in a coordinate system {e k } k∈N that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A (D(B) = D(A)), we give a priori estimates for the asymptotic behavior of the expressions r n = u n − u and R n = Au n − f as n → ∞. A relationship between the order of smallness of these expressions and the degree of smoothness of u with respect to the operator B is established.Let H be a separable Hilbert space over C, let ( · , · ) and · be the corresponding inner product and norm in H, and let A be an invertible closed linear operator with D(A −1 ) = H. (The symbol D( · ) stands for the domain of an operator.)We consider the equation(1) and seek an approximate solution u n by the least squares method, i.e., in the formwhere {e k } k∈N is a given linearly independent system of vectors in D(A) (a so-called coordinate system) and α k ∈ C are some numbers such that R 2 n = Au n − f 2 assumes the least value. The parameters α k are uniquely determined by the system of equations n k=1 α k (Ae k , Ae i ) = (f, Ae i ), i= 1, . . . , n.If the system {e k } k∈N is complete in the Hilbert space H 1 = D(A) with inner product (x, y) 1 = (Ax, Ay), then (e.g., see [1])However, studying the dependence of the asymptotic behavior of r n and R n as n → ∞ on the choice of {e k } k∈N , the properties of the parameters of Eq. (1), and the solution u proves to be a rather difficult problem, which has not yet been solved in the general case. See [2, 3] for some particular results in the case of ordinary differential equations and [4] for Eq. (1) with a positive definite self-adjoint operator A with discrete spectrum. The present paper deals with the solution of this problem for the case in which {e k } k∈N is an orthonormal basis of eigenvectors of a positive definite self-adjoint operator B similar to A, i.e., such that D(B) = D(A). The results presented here are definitive in that the conditions guaranteeing a certain order of smallness of the expressions r n and R n and stated in terms of the degree of smoothness of the solution u with respect to the operator B are necessary and sufficient. We also note that the dependence of the degree of smoothness of u *

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Analytic and entire vectors for representations of Lie groups

Cited by 132 publications

References 22 publications

On extensions and restrictions of semigroups of linear operators in a Banach space and their applications

On extensions and restrictions of semigroups of linear operators in a Banach space and their applications

Analyticité et itères d'un système de champs non elliptique

On the approximation to solutions of operator equations by the least squares method

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