Lately I've been thinking a lot about algebras of operators acting on a Hilbert space, since they provide an extremely useful tool for thinking about locality in quantum field theory. I'm working on a review article about Tomita-Takesaki modular theory to supplement my recent review on the type classification of von Neumann algebras. The core object of study in Tomita-Takesaki theory is a one-parameter group of unitary operators $\Delta^{it},$ generated by a single positive (often unbounded) operator $\Delta.$

In physics, the Tomita-Takesaki unitaries furnish a "hidden thermodynamic symmetry" of a physical state. A lot of interesting physics and mathematics can be learned by studying the analytic structure of the function $z \mapsto \Delta^{z}$ for generic complex $z,$ or of the function $z \mapsto \Delta^{z} |\psi\rangle$ for generic complex $z$ and some fixed state $|\psi\rangle.$ But in order to do this, we need to understand how to do complex analysis for operator-valued (or vector-valued) functions.

In ordinary complex analysis as we learn it in undergrad, a fundamental role is played by Cauchy's two integral theorems, which say that when $f$ is a holomorphic function in a region bounded by a curve $\gamma,$ we have

$$0 = \int_{\gamma} d\zeta\, f(\zeta)$$

and

$$f(z) = \frac{1}{2 \pi i} \int_{\gamma} d\zeta\, \frac{f(\zeta)}{z - \zeta}.$$

Most of the important theorems in complex analysis are proved by rewriting expressions in terms of contour integrals and then manipulating those integrals. To develop a theory of complex analysis for vector- or operator-valued functions, it is therefore essential to begin with a theory of integration for those functions.

This post is about the **Bochner integral**, which is a way of defining integration for functions valued in a Banach space (which could be, for example, a Hilbert space or the space of bounded operators acting on that Hilbert space). The mathematical details of constructing the Bochner integral are interesting, but not by themselves particularly important for physics. What *is *important is understanding: "What manipulations am I allowed to do with a Banach space integral?" The purpose of this post is to outline those rules, and to explain some of their consequences for complex analysis of Banach-valued functions. In a companion post, I will use these ideas to explain **Stone's theorem** in the language of complex analysis. Stone's theorem tells us that every one-parameter group of unitaries on Hilbert space, satisfying an appropriate continuity condition, can be written in the form $t \mapsto \Delta^{it}$ for some operator $\Delta$; it also tells us about the structure of the analytic continuation $\Delta^{z}.$

The outline is:

- In
**section 1**, I will briefly review the essential features of Lebesgue integration; much more detail can be found in my post on $L^p$ spaces. - In
**section 2**, I will define the class of "Bochner measurable" Banach-valued functions on which integration can in-principle be defined, and give a simple criterion for checking whether a Bochner-measurable function has finite integral. I will also show that the Bochner integral is linear and subadditive, and show that continuous functions are always Bochner measurable. - In
**section 3**, I will describe a few important manipulations that can be done with Bochner integrals. In particular, I will explain when a linear operator acting on an integral can be moved inside the integrand, and when the order of an integral and a sum can be interchanged. - In
**section 4**, I will prove some basic theorems of complex analysis for Banach-valued functions.

I learned about the Bochner integral from Wikipedia, Yosida, and Dunford and Schwartz. I learned about Lebesgue integration, which is a prerequisite for the Bochner theory, from Rudin.

**Prerequisites: **Definition of a Banach space; definition of operator norm. Single-variable complex analysis. Comfort with abstract integration and measure spaces.

__Table of Contents__

- Lightning review of Lebesgue integration
- The Bochner integral
- Basic manipulations
- Complex analysis for Banach-valued functions

## 1. Lightning review of Lebesgue integration

**measure space**. Concretely, we specify a

**$\sigma$-algebra**$\Sigma,$ which is a collection of subsets of $\Omega$ that we have declared to be "measurable." We also specify a

**measure**$\mu,$ which is a function $\mu : \Sigma \to [0, \infty]$ that satisfies appropriate additivity criteria. Given a topological space $X$, a function $f : \Omega \to X$ is said to be

**measurable**if $f^{-1}(O)$ is measurable for every open set $O \subseteq X.$

**simple functions**, for which the definition of the integral is obvious. A function $s : \Omega \to \mathbb{C}$ is said to be

**simple**if its range contains only finitely many elements $\{\alpha_1, \dots, \alpha_n\},$ and if for every nonzero $\alpha_j,$ the set $s^{-1}(\alpha_j)$ has finite measure. The integral of $s$ is then naturally defined as

- One shows that every measurable function $f : \Omega \to \mathbb{C}$ can be written as the pointwise limit of a sequence of simple functions $s_n$ with $|s_1| \leq |s_2| \leq \dots \leq |f|.$
- One defines the integral of a measurable function $f : \Omega \to [0, \infty]$ as the limit of $\int d\mu\, s_n$ for a sequence of positive simple functions that approximate $f$ from below; because $\int d\mu\, s_n$ is a monotonically increasing positive sequence, the limit exists (though it may be infinity). One shows that this limit is independent of the sequence of simple functions used to approximate $f$.
- One shows that for any measurable function $f : \Omega \to \mathbb{C},$ if $\int d\mu\, |f|$ is finite, then $\int d\mu\, f$ can be defined uniquely as the linear combination of the integrals of the positive/negative parts of its real/imaginary parts. One also shows that $\int d\mu\, f$ is the limit of $\int d\mu\, s_n$ for a sequence of simple functions approximating $f.$
- One proves various other theorems about the Lebesgue integral: $\int d\mu\, (f + g) = \int d\mu\, f + \int d\mu\, g$ and $|\int d\mu\, f| \leq \int d\mu\, |f|$.

## 2. The Bochner integral

- Any Hilbert space $\mathcal{H}$ with the norm $$\lVert |\psi \rangle \rVert = \sqrt{\langle \psi | \psi \rangle}.$$
- The set of bounded operators on any Hilbert space, $\mathcal{B}(\mathcal{H}),$ with norm

$$\lVert O \rVert_{\infty} = \sup_{\lVert |\psi\rangle \rVert \leq 1} \lVert O |\psi\rangle \rVert.$$ - The complex numbers $\mathbb{C}$ with norm given by the absolute value.

**simple**if its range $\{x_1, \dots, x_n\}$ is finite, and $s^{-1}(x_j)$ has finite measure for any $x_j \neq 0.$ The integral of a simple function is defined as

$$\int d\mu\, s = \sum_{x \in \text{range}(s) - \{0\}} x \cdot \mu(s^{-1}(x)).$$

It is then a straightforward exercise (using basic properties of the measure $\mu$) to show that the integration of simple functions satisfies the important properties of linearity

*not every measurable function $f : \Omega \to X$ can be approximated by simple functions.*This leads us to define a new notion of measurability: a function $f : \Omega \to X$ is said to be

**Bochner measurable**if there exists a sequence of simple functions $s_n : \Omega \to X$ that converges pointwise to $f$ almost everywhere. If there exists such a sequence with the additional property $\int d\mu\, \lVert f - s_n \rVert \to 0,$ then we say $f$ is

**Bochner integrable**, and we define $\int d\mu\, f = \lim_n \int d\mu\, s_n.$

- We need to verify that the function $\lVert f - s_n \rVert$ is measurable, so that it makes sense to ask whether $\int d\mu\, \lVert f - s_n \rVert$ converges to zero.
- We need to verify that when $f$ is Bochner integrable, the limit $\int d\mu\, s_n$ exists.
- We need to verify that this limit is independent of the defining sequence $s_n.$
- We need to verify that this definition of the integral has the crucial properties of linearity and subadditivity.
- We don't
*need*to do this, but it would be nice to have a simpler criterion for integrability than "construct a sequence of simple functions with some special properties."

- First, I will show that if $\Omega$ is a Borel subset of $\mathbb{R}^m$ and $\mu$ is a Borel measure, then any continuous function $f : \Omega \to X$ is Bochner measurable.
- Second, I will show that a Bochner measurable function for any measure space $f : \Omega \to X$ is Bochner integrable iff we have $\int d\mu\, \lVert f \rVert < \infty.$

*and*satisfies $\int d\mu\, \lVert f - r_n \rVert \to 0.$

## 3. Basic manipulations

- If $f : \Omega \to X$ is Bochner integrable and $T : X \to Y$ is linear and bounded, then $T f$ is Bochner integrable and we have $T \int d\mu\, f = \int d\mu\, T f.$
- If $f : \Omega \to X$ is Bochner integrable, $T : X \to Y$ is unbounded but closed (I'll define this later), the image of $f$ lies in the domain of $T$, and $T f$ is Bochner integrable, then we have $T \int d\mu\, f = \int d\mu\, T f.$
- If $\Omega$ is a finite measure space, $f : \Omega \to X$ is Bochner integrable, and $f$ can be written almost everywhere as a series $f(\omega) = \sum_{n} a_n(\omega)$ of Bochner integrable functions such that the series converges absolutely and uniformly, then we have $\int d\mu\, f = \sum_{n} \int d\mu\, a_n.$ (This is a special case of the more general Fubini theorem, but the general case has a less inuitive proof, and we only need the special case for complex analysis.)

**closed operator**is an unbounded operator $T : D_T \to Y$ that has one of the essential properties of bounded operators. If $T$ were bounded, then whenever we have $x_n \to x,$ we have $T x_n \to T x.$ A closed operator has the weaker property that

**if $x_n \to x$ is in $D_T$ and $T x_n$ converges to something, then $T x_n$ must converge to $T x.$**So closed operators need not take all convergent sequences to convergent sequences, but when they do, the image sequence and converges to the image of the limit of the domain sequence. Closed operators are very important in operator theory; for example, every self-adjoint operator on Hilbert space is closed.

**Hille's theorem**. Let us state it again. We will assume that $T : D_T \to Y$ is a closed operator, that $f : \Omega \to X$ is a Bochner integrable function, that the image of $f$ lies in $D_T,$ and that $T f$ is Bochner integrable. We then want to show the identity $T \int d\mu\, f = \int d\mu\, T f.$ The key will be to go to the direct sum space $X \oplus Y.$ Any book on Banach space theory will tell you that $X \oplusY$ is a Banach space with respect to the norm

## 4. Complex analysis for Banach-valued functions

**holomorphic**at the point $z \in \Omega$ if the limit

- If $f : \Omega \to X$ is holomorphic, $\Omega$ is simply connected, and $\gamma$ is a simple closed curve in $\Omega,$ then we have $\int_{\gamma} dz\, f = 0.$
- If $f : \Omega \to X$ is holomorphic and $\Omega$ is simply connected, then for any point $z$ in $\Omega$ and any simple curve $\gamma$ surrounding $z$ with clockwise orientation, we have

$$f(z) = \frac{1}{2 \pi i} \int_{\gamma} dw\, \frac{f(w)}{w - z}.$$ - If $f : \Omega \to X$ is holomorphic, then it is analytic and hence infinitely differentiable.
- If $f : \Omega \to X$ is continuous and $\int_{\gamma} dz\, f(z)$ vanishes for every simple closed curve $\gamma$ in $f,$ then $f$ is holomorphic.
- If $f_n : \bar{\Omega} \to X$ is a sequence of holomorphic functions continuous on the boundary of $\bar{\Omega}$ that converge uniformly to $f : \bar{\Omega} \to X,$ then $f$ is holomorphic in $\Omega$ and continuous on its boundary.

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