A diagonal of a bounded linear operator \(T \in B(\Hil)\) is a sequence of inner products \(\big(\langle Te_n,e_n \rangle\big)\) where \(\{e_n\}_{n=1}^{\infty}\) is an orthonormal basis for the Hilbert space \(\Hil\text{.}\) In other words, a diagonal of \(T\) is the diagonal of some matrix representation of \(T\) with respect to an orthonormal basis.
In his seminal papers on the Pythagorean Theorem [12, 13] Kadison proved the following characterization of diagonals of projections.
A sequence \((d_n)\) is the diagonal of a projection \(P\) if and only if it takes values in the unit interval and the quantities
\begin{equation*} a := \sum_{d_n \lt \frac{1}{2}} d_n \quad\text{and}\quad b := \sum_{d_n \ge \frac{1}{2}} (1-d_n) \end{equation*}satisfy one of the mutually exclusive conditions
The existence of the integer \(a-b\) is not at all obvious and Kadison himself referred to it as “curious.” Since Kadison's initial paper, both Arveson (Theorem 3 of [2]) and Argerami (Theorem 4.6 of [1]) have provided new proofs that \(a-b \in \mathbb{Z}\text{.}\) Recently, the author and Kaftal further clarified this integer in [14] as the essential codimension between the projection \(P\) and a natural diagonal projection associated to \(a,b\text{.}\) Essential codimension was developed by Brown, Douglas and Fillmore in Remark 4.9 of [5] (see also Definition 2.1 below) for pairs of projections whose difference is compact.
Arveson also recognized the Kadison integer as the index of a Fredholm operator in [2], and referred to it as an “index obstruction” to an arbitrary sequence with values in the unit interval being a diagonal of a projection. Arveson was able to extend this index obstruction to any normal operator with finite spectrum coincident with its essential spectrum whose elements are the vertices of a convex polygon. In order to state his main theorem, Arveson associated several objects to a finite set \(X \subseteq \mathbb{C}\text{.}\)
For a finite set \(X \subseteq \mathbb{C}\text{,}\) the sequences which accumulate summably at \(X\) are
\begin{equation*} \Lim (X) := \left\{ (d_n) \in \ell^{\infty} \vmid \sum_{n=1}^{\infty} \dist(d_n,X) \lt \infty \right\}. \end{equation*}For a set \(X = \{\lambda_1,\ldots,\lambda_m\} \subseteq \mathbb{C}\text{,}\) let \(K_X\) denote the \(\mathbb{Z}\)-module of linear combinations over \(\mathbb{Z}\) of elements of \(X\) whose coefficients sum to zero. This can also be expressed as the free \(\mathbb{Z}\)-module generated by \(\lambda_1-\lambda_2, \ldots, \lambda_1-\lambda_m\text{.}\)
For a finite set \(X \subseteq \mathbb{C}\) there is a natural map \(s : \Lim(X) \to \mathbb{C}/K_X\text{.}\) For \((d_n) \in \Lim(X)\text{,}\) since \(X\) is finite there are \(x_n \in X\) for which \(\abs{d_n - x_n} = \dist(d_n,X)\text{,}\) and therefore the series \(\sum_{n=1}^{\infty} (d_n - x_n)\) is absolutely summable. Arveson proved in Proposition 1 of [2] that the coset of this sum in \(\mathbb{C}/K_X\) is independent of the choices of \(x_n \in X\text{,}\) so the map
\begin{equation*} s(d) := \sum_{n=1}^{\infty} (d_n - x_n) + K_X \in \mathbb{C}/K_X \end{equation*}is well-defined. The element \(s(d)\) is called the renormalized sum of \(d\text{.}\)
We reproduce Arveson's theorem (Theorem 4 of [2]) verbatim for reference. Here, \(\mathcal{N}(X)\) denotes the set of normal operators with finite spectrum \(X\) coincident with their essential spectrum.
Let \(X = \{\lambda_1,\ldots,\lambda_m\}\) be the set of vertices of a convex polygon \(P \subseteq \mathbb{C}\) and let \(d = (d_1,d_2,\ldots)\) be a sequence of complex numbers satisfying \(d_n \in P\) for \(n \ge 1\text{,}\) together with the summability condition
\begin{equation} \sum_{n=1}^{\infty} \abs{f(d_n)} \lt \infty,\label{eq-summability-condition}\tag{1.1} \end{equation}where \(f(z) = (z-\lambda_1)(z-\lambda_2)\cdots(z-\lambda_m)\text{.}\) Then \(d \in \Lim(X)\text{;}\) and if \(d\) is the diagonal of an operator \(N \in \mathcal{N}(X)\text{,}\) then \(s(d) = 0\text{.}\)
The summability condition (1.1) is equivalent to \((d_n) \in \Lim(X)\) via a routine analysis argument (see [2] Proposition 2). Moreover, using the notation of Definition 1.4, \(s(d) = 0\) is equivalent by definition to the existence integers \(c_1,\ldots,c_m\) (which depend on the choices \(x_n \in X\)) whose sum is zero for which
\begin{equation} \sum_{n=1}^{\infty} (d_n - x_n) = \sum_{k=1}^m c_k \lambda_k \in K_X.\label{eq-c_k-integers}\tag{1.2} \end{equation}When \(X = \{0,1\} = \spec(N)\text{,}\) \(N\) is a projection, and the condition \((d_n) \in \Lim(\{0,1\})\) is equivalent to \(a+b \lt \infty\text{,}\) where \(a,b\) are defined as in Theorem 1.1. Moreover, \(K_{\{0,1\}} = \mathbb{Z}\text{,}\) so that Arveson's theorem is a generalization of the forward implication Theorem 1.1ii in the situation where \(P\) is an infinite and co-infinite projection.
Our focus is to provide a new approach to Arveson's theorem that, by linking it to the notion of diagonalization by unitaries which are Hilbert–Schmidt perturbations of the identity, permits us both to identify the integers \(c_k\) of (1.2) implicit in the theorem in terms of essential codimension and to eliminate some of the hypotheses in the theorem. Our intent is to bring a fresh perspective on two key parts of Arveson's theorem: the quantity \(\sum_{n=1}^{\infty} (d_n-x_n)\) and the condition \((d_n) \in \Lim \big(\spec(N)\big)\text{.}\) We identify the sum \(\sum_{n=1}^{\infty} (d_n-x_n)\) as \(\trace\big(E(N-N')\big)\) for some diagonal operator \(N'\) with \(\spec(N') \subseteq \spec(N)\) (Proposition 3.5). Here \(E : B(\mathcal{H}) \to \mathcal{A}\) denotes the canonical trace-preserving conditional expectation onto the atomic masa associated to an orthonormal basis; that is, \(E\) is the operation of “taking the main diagonal.” Then we prove that if \(N\) is normal and \(U\) is a unitary which is a Hilbert–Schmidt perturbation of the identity, then \(E(N-UNU^{*})\) is trace-class and \(\trace\big(E(N-UNU^{*})\big) = 0\) (Theorem 3.8). Next, we establish that the condition \((d_n) \in \Lim \big(\spec(N)\big)\) is equivalent to the diagonalizability of \(N\) by a unitary which is a Hilbert–Schmidt perturbation of the identity (Theorem 4.2). The proof relies on essential codimension and a geometric lemma (Lemma 4.1) which is similar to Lemma 1 of [2]. This culminates in a generalization of Arveson's theorem (Theorem 4.3) proved using techniques involving essential codimension, which allows for the identification of the integers \(c_k\) in terms of the essential codimensions of pairs of spectral projections of \(N\) and a diagonal operator \(N'\text{.}\) Finally, we show how our results may be used to derive Arveson's Theorem 1.5.