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 Pythagorean Theorem

(The Standard Generalization)

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

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

Kadison's Observation

\[\|Pe_{1}\|^{2} + \|Pe_{2}\|^{2} = 1\]

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

Similar Triangles!

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

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

Kadison's Observation

\[\|Pe_{1}\|^{2} + \|Pe_{2}\|^{2} = 1\]

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,

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

Kadison's Carpenter's Theorem (2d)

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 \[\|Pe_{i}\|^{2} = d_{i} \quad\text{for}\quad i=1,\ldots,n.\]

Kadison's Carpenter's Theorem

\[\langle Pe_{i},e_{i}\rangle = \]

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

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

Characterization of Diagonals of Projections

in Finite Dimensions

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

self-adjoint finite spectrum

\(\mathrm{II}_{1}\) factors

W/ prescribed singular values

Normal

Two paths forward

?

\(\infty\) 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 \(a-b\in\Z\)

Kadison's Theorem

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

self-adjoint finite spectrum

\(\mathrm{II}_{1}\) factors

W/ prescribed singular values

Normal

Two paths forward

?

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

self-adjoint finite spectrum

\(\mathrm{II}_{1}\) 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

self-adjoint finite spectrum

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

Consider the self-adjoint operator

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

self-adjoint finite spectrum

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

Georgetown Colloquium

By John Jasper

Georgetown Colloquium

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