On complex interpolation and spectral continuity

Karen Saxe

Studia Mathematica (1998)

  • Volume: 130, Issue: 3, page 223-229
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Saxe, Karen. "On complex interpolation and spectral continuity." Studia Mathematica 130.3 (1998): 223-229. <http://eudml.org/doc/216554>.

@article{Saxe1998,
abstract = {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.},
author = {Saxe, Karen},
journal = {Studia Mathematica},
keywords = {complex interpolation; semicontinuous},
language = {eng},
number = {3},
pages = {223-229},
title = {On complex interpolation and spectral continuity},
url = {http://eudml.org/doc/216554},
volume = {130},
year = {1998},
}

TY - JOUR
AU - Saxe, Karen
TI - On complex interpolation and spectral continuity
JO - Studia Mathematica
PY - 1998
VL - 130
IS - 3
SP - 223
EP - 229
AB - 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.
LA - eng
KW - complex interpolation; semicontinuous
UR - http://eudml.org/doc/216554
ER -

References

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  1. [1] J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976. Zbl0344.46071
  2. [2] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. Zbl0204.13703
  3. [3] M. Cwikel, Complex interpolation spaces, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), 1005-1009. Zbl0409.46067
  4. [4] C. J. A. Halberg, The spectra of bounded linear operators on the sequence spaces, Proc. Amer. Math. Soc. 8 (1956), 728-732. Zbl0078.29603
  5. [5] D. A. Herrero and K. Saxe Webb, Spectral continuity in complex interpolation, Math. Balkanica 3 (1989), 325-336. Zbl0708.46060
  6. [6] K. Saxe, Compactness-like operator properties preserved by complex interpolation, Ark. Mat. 35 (1997), 353-362. Zbl0927.46054
  7. [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). 

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