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## Studia Mathematica

1998 | 130 | 3 | 223-229
Tytuł artykułu

### On complex interpolation and spectral continuity

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let $[X_0,X_1]_t$, 0 ≤ t ≤ 1, be Banach spaces obtained via complex interpolation. With suitable hypotheses, linear operators T that act boundedly on both $X_0$ and $X_1$ will act boundedly on each $[X_0,X_1]_t$. Let $T_t$ denote such an operator when considered on $[X_0,X_1]_t$, and $σ(T_t)$ denote its spectrum. We are motivated by the question of whether or not the map $t → σ(T_t)$ is continuous on (0,1); this question remains open. In this paper, we study continuity of two related maps: $t → (σ(T_t))^∧$ (polynomially convex hull) and $t → ∂_e(σ(T_t))$ (boundary of the polynomially convex hull). We show that the first of these maps is always upper semicontinuous, and the second is always lower semicontinuous. Using an example from [5], we now have definitive information: $t → (σ(T_t))^∧$ is upper semicontinuous but not necessarily continuous, and $t → ∂_e(σ(T_t))$ is lower semicontinuous but not necessarily continuous.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
223-229
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-02-12
poprawiono
1998-01-12
Twórcy
autor
• Department of Mathematicsi, Macalester College, 1600 Grand Avenue, St. Paul, Minnesota 55105 U.S.A. , saxe@macalester.edu
Bibliografia
• [1] J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976.
• [2] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190.
• [3] M. Cwikel, Complex interpolation spaces, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), 1005-1009.
• [4] C. J. A. Halberg, The spectra of bounded linear operators on the sequence spaces, Proc. Amer. Math. Soc. 8 (1956), 728-732.
• [5] D. A. Herrero and K. Saxe Webb, Spectral continuity in complex interpolation, Math. Balkanica 3 (1989), 325-336.
• [6] K. Saxe, Compactness-like operator properties preserved by complex interpolation, Ark. Mat. 35 (1997), 353-362.
• [7] I. Ja. Šneĭberg [I. Ya. Shneĭberg], Spectral properties of linear operators in interpolation families of Banach spaces, Mat. Issled. 9 (1974), no. 2, 214-229 (in Russian).
Typ dokumentu
Bibliografia
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