# Interpolation of Holomorphic functions

## TL;DR

This paper explores conditions under which holomorphic functions maintain their compactness and holomorphic properties when restricted to interpolated Banach spaces, advancing interpolation theory for complex analysis.

## Contribution

It provides new criteria ensuring that restrictions of holomorphic maps to interpolated spaces are both compact and holomorphic, extending existing interpolation results.

## Key findings

- Identifies conditions for compactness preservation in holomorphic interpolations
- Establishes when holomorphicity is retained after interpolation
- Enhances understanding of interpolation in complex Banach spaces

## Abstract

Interpolation Theory gives techniques for constructing spaces from two initial Banach spaces. We provide several conditions under which the restriction of a holomorphic map $f:X_0+X_1 \rightarrow Y_0+Y_1$ to the interpolated spaces (using some specific interpolation methods), where $f_{|X_0}:X_0 \rightarrow Y_0$ is compact, is also compact and holomorphic.