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

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!

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

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

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

\(\Box\)

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. 

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

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

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

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

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 

Projections in infinite dimensions

Examples. 

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

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

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)

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

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

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 \[c-d=N\alpha+k\quad\text{and}\quad c\geq (N+k)\alpha.\]

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

An application

Consider the self-adjoint operator

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?