# Diagonals of Operators

John Jasper

South Dakota State University

### The Pythagorean Theorem

Theorem. If $$\Delta$$ is a right triangle with side lengths $$c\geq b\geq a$$, then $a^{2}+b^{2}=c^{2}.$

$$a$$

$$b$$

$$c$$

### The Pythagorean Theorem

Theorem. If $$v$$ and $$w$$ are orthogonal vectors, then $\|v\|^2 + \|w\|^2 = \|v+w\|^2.$

$$v$$

$$w$$

$$v+w$$

### (The Standard Generalization)

Theorem. If $$v_{1},v_{2},\ldots,v_{k}$$ are pairwise orthogonal vectors, then

$\|v_{1}\|^2 + \|v_{2}\|^2 + \cdots + \|v_{k}\|^{2} = \|v_{1}+v_{2}+\cdots+v_{k}\|^2.$

$\|Pe_{1}\|^{2} + \|Pe_{2}\|^{2} = 1$

Similar Triangles!

$\|(I-P)e_{2}\| = \|Pe_{1}\|$

$\|Pe_{1}\|^{2} + \|Pe_{2}\|^{2} = 1$

### The Pythagorean Theorem

Theorem. If $$P$$ is an orthogonal projection onto a $$1$$-dimensional subspace $$V$$, and $$\{e_{1},e_{2},\ldots,e_{n}\}$$ is an orthonormal basis, then

$\sum_{i=1}^{n}\|Pe_{i}\|^{2} = 1.$

Proof.

• Let $$v\in V$$ be a unit vector.
• $$Px = \langle x,v\rangle v,$$
• $$\|Pe_{i}\|^{2} = |\langle e_{i},v\rangle|^{2}$$
• $$\displaystyle{\sum_{i=1}^{n}\|Pe_{i}\|^{2} = \sum_{i=1}^{n}|\langle e_{i},v\rangle|^{2} = \|v\|^{2} = 1}.$$

$$\Box$$

### The Pythagorean Theorem and Diagonals

If $$P$$ is an orthogonal projection onto a subspace $$V$$, and $$(e_{i})_{i=1}^{n}$$ is an orthonormal basis, then

$\|Pe_{i}\|^{2} = \langle Pe_{i},Pe_{i}\rangle = \langle P^{\ast}Pe_{i},e_{i}\rangle = \langle P^{2}e_{i},e_{i}\rangle = \langle Pe_{i},e_{i}\rangle$

$= \left[\begin{array}{cccc} \|Pe_{1}\|^{2} & \ast & \cdots & \ast\\ \overline{\ast} & \|Pe_{2}\|^{2} & & \vdots\\ \vdots & & \ddots & \vdots\\ \overline{\ast} & \cdots & \cdots & \|Pe_{n}\|^{2}\end{array}\right]$

$P= \left[\begin{array}{cccc} \langle Pe_{1},e_{1}\rangle & \langle Pe_{2},e_{1}\rangle & \cdots & \langle Pe_{n},e_{1}\rangle \\ \langle Pe_{1},e_{2}\rangle & \langle Pe_{2},e_{2}\rangle & & \vdots\\ \vdots & & \ddots & \vdots\\ \langle Pe_{1},e_{n}\rangle & \cdots & \cdots & \langle Pe_{n},e_{n}\rangle \end{array}\right]$

### The Pythagorean Theorem

Theorem. If $$P$$ is an orthogonal projection onto a $$1$$-dimensional subspace $$V$$, and $$\{e_{1},e_{2},\ldots,e_{n}\}$$ is an orthonormal basis, then

$\sum_{i=1}^{n}\|Pe_{i}\|^{2} = 1.$

Proof. $\sum_{i=1}^{n}\|Pe_{i}\|^{2} = \text{tr}(P) = \dim V = 1.$

$$\Box$$

### The Pythagorean Theorem

Theorem. If $$P$$ is an orthogonal projection onto a $$k$$-dimensional subspace $$V$$, and $$\{e_{1},e_{2},\ldots,e_{n}\}$$ is an orthonormal basis, then

$\sum_{i=1}^{n}\|Pe_{i}\|^{2} = k.$

Proof. $\sum_{i=1}^{n}\|Pe_{i}\|^{2} = \text{tr}(P) = \dim V = k.$

$$\Box$$

### Kadison's Pythagorean Theorem (finite dimensional)

Theorem. If $$P$$ is an orthogonal projection onto a $$k$$-dimensional subspace $$V$$, and $$\{e_{1},e_{2},\ldots,e_{n}\}$$ is an orthonormal basis, then

$\sum_{i=1}^{n}\|Pe_{i}\|^{2} = k.$

Proof. $\sum_{i=1}^{n}\|Pe_{i}\|^{2} = \text{tr}(P) = \dim V = k.$

$$\Box$$

Corollary. If $$P$$ is an orthogonal projection onto a $$k$$-dimensional subspace, and $$(d_{i})_{i=1}^{n}$$ is the sequence on the diagonal of $$P$$, then

$$d_{i}\in[0,1]$$ for each $$i$$, and

$\sum_{i=1}^{n}d_{i} \in\Z$

### The Carpenter's Theorem

Theorem. If $$\Delta$$ is a triangle with side lengths $$c\geq b\geq a$$, such that

$$a^{2}+b^{2}=c^{2},$$ then $$\Delta$$ is a right triangle.

$$a$$

$$b$$

$$c$$

Proof. Law of cosines

$c^2=a^2+b^2-2ab\cos(\theta).$

$$\theta$$

Theorem. If $$v$$ and $$w$$ are vectors in a real Hilbert space such that

$$\|v\|^2 + \|w\|^2 = \|v+w\|^2,$$ then $$\langle v,w\rangle = 0.$$

### The Carpenter's Theorem

Proof.

$\|v+w\|^2 = \langle v+w,v+w\rangle = \|v\|^2+2\langle v,w\rangle + \|w\|^{2}$

$\|v+w\|^2=\|v\|^2+\|w\|^2 \quad \Rightarrow\quad 2\langle v,w\rangle = 0. \quad \Box$

Theorem. If $$d_{1},d_{2}$$ are two numbers in $$[0,1]$$ such that $$d_{1}+d_{2} = 1,$$ then there is a projection $$P$$ such that $$d_{1} = \|Pe_{1}\|^2$$ and $$d_{2} = \|Pe_{2}\|^2$$, that is,

$P = \begin{bmatrix} d_{1} & \alpha\\ \overline{\alpha} & d_{2}\end{bmatrix}.$

Proof.

• By the intermediate value theorem there is a projection $$P$$ such that $$\|Pe_{1}\|^{2}=d_{1}.$$
• By the Pythagorean theorem                $$\|Pe_{1}\|^2+\|Pe_{2}\|^2=1.$$
• Therefore, $\|Pe_{2}\|^2=d_{2}.$

Theorem. If $$(d_{i})_{i=1}^{n}$$ is a sequence of numbers in $$[0,1]$$ such that $\sum_{i=1}^{n}d_{i}\in\N\cup\{0\},$ then there is an $$n\times n$$ projection $$P$$ such that $\|Pe_{i}\|^{2} = d_{i} \quad\text{for}\quad i=1,\ldots,n.$

$\langle Pe_{i},e_{i}\rangle =$

Note that this means that the sequence on the diagonal of the matrix $$P$$ is $$(d_{i})_{i=1}^{n}$$.

Example. Consider the sequence

$\left(\frac{5}{7},\frac{5}{7},\frac{3}{7},\frac{1}{7}\right).$

$\left[\begin{array}{rrrr}\frac{5}{7} & -\frac{\sqrt{15}}{21} & -\frac{\sqrt{30}}{21} & \frac{\sqrt{5}}{7}\\[1ex] -\frac{\sqrt{15}}{21} & \frac{5}{7} & -\frac{2\sqrt{2}}{7} & -\frac{\sqrt{3}}{21}\\[1ex] -\frac{\sqrt{30}}{21} & -\frac{2\sqrt{2}}{7} & \frac{3}{7} & -\frac{\sqrt{6}}{21}\\[1ex] \frac{\sqrt{5}}{7} & -\frac{\sqrt{3}}{21} & -\frac{\sqrt{6}}{21} & \frac{1}{7}\end{array}\right]$

Challenge: Construct a $$4\times 4$$ projection with this diagonal.

Theorem. Assume $$(d_{i})_{i=1}^{n}$$ is a sequence of numbers in $$[0,1].$$ There is an $$n\times n$$ projection $$P$$ with diagonal $$(d_{i})_{i=1}^{n}$$ if and only if

$\sum_{i=1}^{n}d_{i} \in\N\cup\{0\}.$

### Diagonality

Definition. Given an operator $$E$$ on a Hilbert space, a sequence $$(d_{i})_{i\in I}$$ is a diagonal of $$E$$ if there is an orthonormal basis $$(e_{i})_{i\in I}$$ such that

$d_{i} = \langle Ee_{i},e_{i}\rangle \quad\text{for all }i\in I.$

The problem: Given an operator $$E$$, characterize the set of diagonals of $$E$$, that is, the set

$\big\{(\langle Ee_{i},e_{i}\rangle )_{i\in I} : (e_{i})_{i\in I}\text{ is an orthonormal basis}\big\}$

In particular, we want a characterization in terms of linear inequalities between the diagonal sequences and the spectral information of $$E$$.

Diagonals of projections in finite dimensions

Diagonals of projections in infinite dimensions

Diagonals of self-adjoint matrices in finite dimensions

Compact positive

Normal

$$\mathrm{II}_{1}$$ factors

W/ prescribed singular values

Normal

### ?

$$\infty$$ dimensional path

Finite dimensional path

### Projections in infinite dimensions

Examples. Let $$(e_{i})_{i=1}^{\infty}$$ be an orthonormal basis.

Set $$\displaystyle{v = \sum_{i=1}^{\infty}\sqrt{\frac{1}{2^{i}}}e_{i}},$$ then

$I-P = \begin{bmatrix} \frac{1}{2} & -\frac{1}{2^{3/2}} & -\frac{1}{2^{2}} & \cdots \\[1ex] -\frac{1}{2^{3/2}} & \frac{3}{4} & -\frac{1}{2^{5/2}} & \cdots\\[1ex] -\frac{1}{2^{2}} & -\frac{1}{2^{5/2}} & \frac{7}{8} & \cdots\\ \vdots & \vdots & \vdots & \ddots\end{bmatrix}$

$P = \langle \cdot,v\rangle v = \begin{bmatrix} \frac{1}{2} & \frac{1}{2^{3/2}} & \frac{1}{2^{2}} & \cdots \\[1ex] \frac{1}{2^{3/2}} & \frac{1}{4} & \frac{1}{2^{5/2}} & \cdots\\[1ex] \frac{1}{2^{2}} & \frac{1}{2^{5/2}} & \frac{1}{8} & \cdots\\[1ex] \vdots & \vdots & \vdots & \ddots\end{bmatrix}$

Corank 1 projection

Diagonal: $$\displaystyle{\left(\frac{1}{2},\frac{3}{4},\frac{7}{8},\ldots\right)}$$

Rank 1 projection

Diagonal: $$\displaystyle{\left(\frac{1}{2},\frac{1}{4},\frac{1}{8},\ldots\right)}$$

### Projections in infinite dimensions

Examples.

$\frac{1}{2}J_{2} = \begin{bmatrix} \frac{1}{2} & \frac{1}{2}\\[1ex] \frac{1}{2} & \frac{1}{2}\end{bmatrix}$

$Q = \bigoplus_{i=1}^{\infty}\frac{1}{2}J_{2} = \begin{bmatrix} \frac{1}{2}J_{2} & \mathbf{0} & \mathbf{0} & \cdots\\ \mathbf{0} & \frac{1}{2}J_{2} & \mathbf{0} & \cdots\\ \mathbf{0} & \mathbf{0} & \frac{1}{2}J_{2} & \\ \vdots & \vdots & & \ddots\end{bmatrix}$

$$\infty$$-rank and $$\infty$$-corank

Diagonal: $$\displaystyle{\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\ldots\right)}$$

$$\infty$$-rank and $$\infty$$-corank

Diagonal: $$\displaystyle{\left(\ldots,\frac{1}{8},\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4},\frac{7}{8},\ldots\right)}$$

$P\oplus (I-P)$

Theorem. Assume $$(d_{i})_{i=1}^{n}$$ is a sequence of numbers in $$[0,1].$$ There is an $$n\times n$$ projection $$P$$ with diagonal $$(d_{i})_{i=1}^{n}$$ if and only if

$\sum_{i=1}^{n}d_{i} \in\N\cup\{0\}.$

### Diagonals of Projections Redux

$\sum_{i=1}^{k}d_{i} - \sum_{i=k+1}^{n}(1-d_{i})\in\Z$

$\Updownarrow$

Theorem. Assume $$(d_{i})_{i=1}^{n}$$ is a sequence of numbers in $$[0,1].$$ There is an $$n\times n$$ projection $$P$$ with diagonal $$(d_{i})_{i=1}^{n}$$ if and only if

$\sum_{i=1}^{k}d_{i} - \sum_{i=k+1}^{n}(1-d_{i})\in\Z.$

### Diagonals of Projections Redux

Theorem (Kadison '02). Assume $$(d_{i})_{i=1}^{\infty}$$ is a sequence of numbers in $$[0,1],$$ and set

$a=\sum_{d_{i}<\frac{1}{2}}d_{i}\quad\text{and}\quad b=\sum_{d_{i}\geq \frac{1}{2}}(1-d_{i})$

There is a projection $$P$$ with diagonal $$(d_{i})_{i=1}^{\infty}$$ if and only if one of the following holds:

1. $$a=\infty$$
2. $$b=\infty$$
3. $$a,b<\infty$$ and $$a-b\in\Z$$

Examples.

• There is a projection with every number in $$\mathbb{Q}\cap[0,1]$$ on the diagonal.
• There is a projection with diagonal $$(\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{4},\ldots)$$.
• There is no projection with diagonal $\left(\ldots,\frac{1}{25},\frac{1}{16},\frac{1}{9},\frac{1}{4},\frac{1}{2},\frac{3}{4},\frac{7}{8},\frac{15}{16},\ldots\right)$

Diagonals of projections in finite dimensions

Diagonals of projections in infinite dimensions

Diagonals of self-adjoint matrices in finite dimensions

Compact positive

Normal

$$\mathrm{II}_{1}$$ factors

W/ prescribed singular values

Normal

### The Schur-Horn Theorem

Theorem (Schur '23, Horn '54). Let $$(d_{i})_{i=1}^{n}$$ and $$(\lambda_{i})_{i=1}^{n}$$ be nonincreasing sequences. There is a self-adjoint matrix $$E$$ with diagonal $$(d_{i})_{i=1}^{n}$$ and eigenvalues $$(\lambda_{i})_{i=1}^{n}$$ if and only if

$\sum_{i=1}^{k}d_{i}\leq \sum_{i=1}^{k}\lambda_{i}\quad\text{for}\quad k=1,2,\ldots,n-1$

and

$\sum_{i=1}^{n}d_{i} = \sum_{i=1}^{n}\lambda_{i}.$

(1)

(2)

If (1) and (2) hold, then we say that $$(\lambda_{i})_{i=1}^{n}$$ majorizes $$(d_{i})_{i=1}^{n}$$, and we write $$(\lambda_{i})_{i=1}^{n}\succeq (d_{i})_{i=1}^{n}$$

$$(\lambda_{i})_{i=1}^{n}\succeq (d_{i})_{i=1}^{n}$$ is equivalent to saying that $$(d_{i})_{i=1}^{n}$$ is in the convex hull of the permutations of $$(\lambda_{i})_{i=1}^{n}$$.

Diagonals of projections in finite dimensions

Diagonals of projections in infinite dimensions

Diagonals of self-adjoint matrices in finite dimensions

Compact positive

Normal

$$\mathrm{II}_{1}$$ factors

W/ prescribed singular values

Normal

### Two paths forward

?

Theorem (Arveson, Kadison '06, Kaftal, Weiss '10). Let $$(\lambda_{i})_{i=1}^{\infty}$$ be a positive nonincreasing sequence, and let $$(d_{i})_{i=1}^{\infty}$$ be a nonnegative nonincreasing sequence. There exists a positive compact operator with diagonal $$(d_{i})_{i=1}^{\infty}$$ and whose positive eigenvalues are $$(\lambda_{i})_{i=1}^{\infty}$$ if and only if

$\sum_{i=1}^{k}d_{i}\leq \sum_{i=1}^{k}\lambda_{i}\quad\text{for all}\quad k\in\N$

and

$\sum_{i=1}^{\infty}d_{i} = \sum_{i=1}^{\infty}\lambda_{i}.$

Open question: What are the diagonals of positive compact operators with positive eigenvalues

$\left(1,\frac{1}{2},\frac{1}{3},\ldots\right)$ and a $$1$$-dimensional kernel.

Diagonals of projections in finite dimensions

Diagonals of projections in infinite dimensions

Diagonals of self-adjoint matrices in finite dimensions

Compact positive

Normal

W/ prescribed singular values

Normal

### Two paths forward

?

$$\mathrm{II}_{1}$$ factors

?

### Three Point Spectrum

Theorem (JJ '13). Let $$0<\alpha<1$$ and let $$(d_{i})_{i=1}^{\infty}$$ be a sequence in $$[0,1]$$. Define

$c=\sum_{d_{i}<\alpha}d_{i}\quad\text{and}\quad d=\sum_{d_{i}\geq \alpha}(1-d_{i}).$

There is a self-adjoint operator $$E$$ with diagonal $$(d_{i})_{i=1}^{\infty}$$ and spectrum $$\{0,\alpha,1\}$$ if and only if one of the following holds:

• $$c=\infty$$
• $$d=\infty$$
• $$c,d<\infty$$ and there exist $$N\in\N$$ and $$k\in\Z$$ such that $c-d=N\alpha+k\quad\text{and}\quad c\geq (N+k)\alpha.$

Notes:

• There is a longer version where the multiplicities of the eigenvalues are considered.
• Together with Marcin Bownik we extended this to all self-adjoint operators with a finite spectrum.

$E=\left(\begin{array}{ccccccc} 1/2 & \overline{\ast} & \overline{\ast} & \overline{\ast} & \overline{\ast} & \cdots\\ \ast & 1/4 & \overline{\ast} & \overline{\ast} & \overline{\ast} & \cdots\\ \ast & \ast & 3/4 & \overline{\ast} & \overline{\ast} & \cdots\\ \ast & \ast & \ast & 1/8 & \overline{\ast} & \cdots\\ \ast & \ast & \ast & \ast & 7/8 & \\ \vdots & \vdots & \vdots & \vdots & & \ddots\\ \end{array}\right)$

### An application

If the eigenvalues of $$E$$ are $$\{0,\alpha,1\}$$ with $$0<\alpha<1$$, then what are the possible values of $$\alpha$$?

$\frac{1}{16},\frac{1}{14},\frac{1}{12},\frac{1}{10},\frac{1}{8},\frac{1}{6},\frac{1}{4},\frac{1}{2},\frac{3}{4},\frac{5}{6},\frac{7}{8},\frac{9}{10},\frac{11}{12},\frac{13}{14},\frac{15}{16}$

Diagonals of projections in finite dimensions

Diagonals of projections in infinite dimensions

Diagonals of self-adjoint matrices in finite dimensions

Compact positive

Normal

W/ prescribed singular values

Normal

### What's left?

?

$$\mathrm{II}_{1}$$ factors

Infinite dimensions:

1. Compact positive operators: What about finite dimensional kernels?
2. Normal operators: Arveson proved a generalization of Kadison's Pythagorean theorem that applies to very special normal operators.
3. $$\mathrm{II}_{1}$$ factors: Ravichandran announced a solution in 2012, but it has not appeared in print.
4. and much more...

### What's left?

Finite dimensions:

1. Normal operators: Diagonals of $$3\times 3$$ normal matrices was worked out by Williams in 1971, but the $$4\times 4$$ case is still open.
2. Matrices with prescribed singular values: Thompson 1977

By John Jasper

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