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## Studia Mathematica

1998 | 128 | 2 | 135-144
Tytuł artykułu

### Sur les isométries partielles maximales essentielles

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EN
We study the problem of approximation by the sets S + K(H), $S_e$, V + K(H) and $V_e$ where H is a separable complex Hilbert space, K(H) is the ideal of compact operators, $S = {L ∈ B(H) : L*L = I}$ is the set of isometries, V = S ∪ S* is the set of maximal partial isometries, $S_e = {L ∈ B(H): π(L*)π( L) = π(I)}$ and $V_e = S_e ∪ S_e*$ where π : B(H) → B(H)/K(H) denotes the canonical projection. We also prove that all the relevant distances are attained. This implies that all these classes are closed and we remark that $V_e = V + K(H)$. We also show that S + K(H) is both closed and open in $S_e$. Finally, we prove that $V_e$, S + K(H) and $S_e$ coincide with their boundaries $∂(V_e)$, ∂(S + K(H)) and $∂(S_e)$ respectively.
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Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
135-144
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-10-15
poprawiono
1997-09-12
Twórcy
autor
• URA au CNRS et UFR de Mathématiques, Bât. M2, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d'Ascq Cedex, France
Bibliografia
• [1] C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), 279-294.
• [2] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), 513-517.
• [3] R. Bouldin, Approximation by semi-Fredholm operators with fixed nullity, Rocky Mountain J. Math. 20 (1990), 39-50.
• [4] J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990.
• [5] R. S. Doran and V. A. Belfi, Characterizations of C*-algebras. The Gelfand-Naimark Theorems, M. Dekker, New York, 1986.
• [6] P. A. Fillmore, J. G. Stampfli and J. P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179-192.
• [7] S. Goldberg, Unbounded Linear Operators, McGraw-Hill, New York, 1966.
• [8] P. R. Halmos, A Hilbert Space Problem Book, D. Van Nostrand, 1967.
• [9] P. de la Harpe, Initiation à l'algèbre de Calkin, Lecture Notes in Math. 725, Springer, 1978, 180-219.
• [10] D. A. Herrero, Approximation of Hilbert Space Operators, Vol. I, Pitman, Boston, 1982.
• [11] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
• [12] D. D. Rogers, Approximation by unitary and essentially unitary operators, Acta Sci. Math. (Szeged) 39 (1977), 141-151.
• [13] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980.
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