Any time you can order mathematical objects, it is productive to ask what operations preserve the ordering. For example, real numbers have a natural ordering, and we have $x \geq y \Rightarrow x^k \geq y^k$ for any odd natural number $k$. If we further impose the assumption $y \geq 0,$ then order preservation holds for $k$ any positive real number.

Self-adjoint operators on a Hilbert space have a natural (partial) order as well. We write $A \geq 0$ for a self-adjoint operator $A$ if we have

$$\langle \psi | A | \psi \rangle \geq 0$$

for every vector $|\psi\rangle,$ and we write $A \geq B$ for self-adjoint operators $A$ and $B$ if we have $(A - B) \geq 0.$ Curiously, many operations that are monotonic for real numbers are not monotonic for matrices. For example, the matrices

$$P = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$

and

$$Q = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

are both self-adjoint and positive, so we have $P+Q \geq P \geq 0$, but a straightforward calculation shows that $(P+Q)^2 - P^2$ is not a positive matrix.

There is a very beautiful theory due to Charles Loewner that explains exactly when a function of matrices preserves order. For any function $f : \mathbb{R} \to \mathbb{R},$ we can define a corresponding function acting on Hermitian matrices by diagonalizing and acting on each eigenvalue individually. Loewner proved first that the only functions that are order-preserving for every pair of Hermitian matrices are the affine functions $f(x) = \alpha x + \beta.$ On its own this is a bit disappointing, but Loewner went on to observe that for there are additional functions that are order-preserving for certain restricted families of Hermitian matrices. In particular, he considered the family of functions that is order-preserving for all matrices with eigenvalues contained in some fixed subset of the real line. For example, there are many more functions that satisfy $A \geq B > 0 \Rightarrow f(A) \geq f(B)$ then there are functions that satisfy the same implication without the requirement that $A$ and $B$ are individually positive. Quite generally, Loewner showed the following:

Let $(a,b)$ be an open subset of $\mathbb{R}$. Then in order to have $f(A) \geq f(B)$ for every $A \geq B$ with eigenvalues of $A$ and $B$ both contained in $(a, b)$, it is necessary and sufficient that $f : (a, b) \to \mathbb{R}$ be a **Pick function**. The function $f : (a, b) \to \mathbb{R}$ is a Pick function if it is real analytic and has an analytic continuation that maps the upper half-plane to itself.

This is a very interesting theorem, as it relates an algebraic structure (matrix order) to an analytic one (holomorphy). The "necessary" part of Loewner's theorem is quite hard to show; this is the part that says any matrix monotone function has an appropriate analytic continuation. But the "sufficient" part is usually what is needed in practice, and the proof is both digestible and instructive. In this post, we will (i) explore the structure of Pick functions and show that they always have interesting integral representations, (ii) use these integral representations to show that Pick functions are monotones for bounded Hermitian operators (going beyond the special case of finite-dimensional matrices), (iii) introduce and study an order relation for unbounded operators, and (iv) explore when Pick functions are monotone for unbounded operators.

I learned this material by studying Schmudgen's book Unbounded self-adjoint operators on Hilbert space and Simon's book Loewner's theorem on monotone matrix functions.

**Prerequisites:** For section 1, complex analysis. For section 2, the spectrum of an operator in terms of its resolvents and the spectral theorem in terms of spectral measures. For sections 3 and 4, basic definitions and propositions concerning unbounded operators.

__Table of Contents__

- Pick functions and integral representations
- Pick functions as bounded monotones
- Order relations for unbounded operators
- Pick functions as unbounded monotones

## 1. Pick functions and integral representations

**Pick function**for a holomorphic map from the upper half-plane to itself: $f : \mathbb{C}^+ \to \mathbb{C}^+.$ We will later consider special cases where $f$ has a real analytic limit on some subset of the real line.

*imaginary part*of $f_{\text{disk}}$ in terms of itself; the real part of the function does not appear anywhere. We can compose this function with a Mobius transformation and change variables to obtain

## 2. Pick functions as bounded monotones

## 3. Order relations for unbounded operators

*bounded below*operators, there is one. Let us first restrict to positive self-adjoint operators. What does it mean to say $A \geq B \geq 0$? First, note that the expression $\langle \psi | A | \psi \rangle$ makes sense not just for every $|\psi \rangle$ in the domain of $A$, but for every $|\psi\rangle$ in the (generally larger) domain of $A^{1/2},$ since we can consider the quantity $\langle A^{1/2} \psi | A^{1/2} \psi \rangle.$ We would like $A \geq B \geq 0$ to be some statement like

*single*$s > c$ to know that it is true for all $s > c$ and also that we have $A \geq B \geq -c$ in the domain sense explained above.

## 4. Pick functions as unbounded monotones

*sort of*. We've only defined an order for bounded-below unbounded operators; consequently, we should only expect to be able to talk about Pick functions being monotonic on unbounded operators when the images of the bounded-below operators under Pick functions are also bounded below. We'll go a little beyond this at the end, but let's start with the simpler case first.

*is*true is that for any $|\psi\rangle$ that is known to be in the simultaneous domain of both, one can show

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