Banach-space-valued analytic functions

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Problem VII.3.4 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section VII.3 "The Spectrum"

If $G$ is an open subset of $\mathbb {C}$ and $f: G \to \mathcal{X}$ is a function such that for each $x^*$ in $\mathcal{X}^*$ , $x^* \circ f: G \to \mathbb {C}$ is analytic, then $f$ is analytic. If the word continuous'' is substituted for both occurrences of the word analytic,'' is the preceding statement still true?

Problem 3.3.26 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.3 "Convexity"

(a)(b)(c)

Suppose $\Omega$ is a region in $\mathbb {C}, X$ is a Fréchet space, and $f: \Omega \rightarrow X$ is holomorphic.

(a)

State and prove a theorem concerning the power series representation of $f$ , that is, concerning the formula $f(z) = \sum (z-a)^{n} c_{n}$ , where $c_{n} \in X$ .

(b)

Generalize Morera's theorem to $X$ -valued holomorphic functions.

(c)

For a sequence of complex holomorphic functions in $\Omega$ , uniform convergence on compact subsets of $\Omega$ implies that the limit is holomorphic. Does this generalize to $X$ -valued holomorphic functions?

Problem 3.10.9 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.10 "Banach Algebras"

Suppose that $\Omega$ is open in $C$ and that $f: \Omega \rightarrow A$ and $\varphi : \Omega \rightarrow C$ are holomorphic. Prove that $\varphi f: \Omega \rightarrow A$ is holomorphic.

Remark.

This was used in the proof of Theorem 10.13, with $\varphi (\lambda ) = \lambda^{n}$ .

Problem 3.10.18 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.10 "Banach Algebras"

(a)(b)

Suppose $\Omega$ is open in $\mathcal{C}, \alpha$ is an isolated boundary point of $\Omega , f: \Omega \rightarrow X$ is a holomorphic $X$ -valued function (where $X$ is some complex Banach space), and there is a smallest nonnegative integer $n$ such that

\left|\lambda -\alpha \right|^{n}\left\| f(\lambda )\right\|

is bounded as $\lambda \rightarrow \alpha$ . If $n > 0$ , then $f$ is said to have a pole of order $n$ at $\alpha$ .

(a)

Suppose $x \in A$ and $(\lambda e-x)^{-1}$ has a pole at every point of $\sigma (x)$ . Prove that there is a nontrivial polynomial $P$ such that $P(x) = 0$ .

(b)

As a special case of $(a)$ , assume $\sigma (x) = \left\{ 0\right\}$ and $(\lambda e-x)^{-1}$ has a pole of order $n$ at 0. Prove that $x^{n} = 0$ .

Remark.

In part (a): this can happen only when $\sigma (x)$ is a finite set.

References

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.