On complex interpolation and spectral continuity

• Autorzy: Karen Saxe
• 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.

Kategorie tematyczne

• 47Bxx: Special classes of linear operators
• 46B70: Interpolation between normed linear spaces
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 130
• Numer: 3
• Strony: 223-229

Daty

wydano

1998

otrzymano

1997-02-12

poprawiono

1998-01-12

Twórcy

autor

Karen Saxe

• Department of Mathematicsi, Macalester College, 1600 Grand Avenue, St. Paul, Minnesota 55105 U.S.A.

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).