# On complex interpolation and spectral continuity

Studia Mathematica (1998)

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

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topSaxe, 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

top- [1] J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976. Zbl0344.46071
- [2] A. P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. Zbl0204.13703
- [3] M. Cwikel, Complex interpolation spaces, a discrete definition and reiteration, Indiana Univ. Math. J. 27 (1978), 1005-1009. Zbl0409.46067
- [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] D. A. Herrero and K. Saxe Webb, Spectral continuity in complex interpolation, Math. Balkanica 3 (1989), 325-336. Zbl0708.46060
- [6] K. Saxe, Compactness-like operator properties preserved by complex interpolation, Ark. Mat. 35 (1997), 353-362. Zbl0927.46054
- [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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