In his seminal papers on the Pythagorean Theorem ([17], [18]), Kadison characterizes the diagonals of projections, that is the sequences that can appear on the diagonal of a matrix representation of a projection. The main assertion of his Theorem 15 is by now usually paraphrased as follows:

Kadison proved that \(a-b\) is arbitrarily close to an integer and hence is an integer and referred to that integer as “curious”.

Let us first express \(a\) and \(b\) in operator theoretic terms. Call \(\projone{}\) the projection, \(\{e_j\}\) the orthonormal basis of \(\h\) used for the matrix representation, and \(\projtwo{}\) the projection on \(\overline {\spans}\{e_j\mid\ d_j> 1/2 \}.\) Then

hence if \(a+b\lt\infty\text{,}\) we have \(\projtwo{}^\perp (\projone{}-\projtwo{}) \projtwo{}^\perp=\projtwo{}^\perp \projone{} \projtwo{}^\perp \in \mathcal L^1\text{,}\) \(\projtwo{}(\projone{}-\projtwo{})\projtwo{}=-(\projtwo{}- \projtwo{}\projone{}\projtwo{}) \in \mathcal L^1\) (where \(\mathcal{L}^1\) denotes the ideal of trace-class operators), and

If we knew that \(\projone{}-\projtwo{}\in \mathcal L^1\text{,}\) then we would have \(a-b=\tr(\projone{}-\projtwo{})\) and then, by [15], we could conclude that \(\tr(\projone{}-\projtwo{})\in\mathbb Z\text{.}\) However, since \(\projone{}-\projtwo{}\) is not necessarily positive, the fact that its corners are trace-class does not imply that \(\projone{}-\projtwo{}\) itself is trace-class. In fact, Argerami proved in [2] that \(\projone{}-\projtwo{}\in \mathcal L^2\) (where \(\mathcal{L}^2\) is the ideal of Hilbert–Schmidt operators) and by modifying Effros' argument, he showed that this is sufficient to guarantee that \(a-b\) is an integer. However, neither Kadison's nor Argerami's proof shed much light on the origin of that integer itself.

One of Bill Arveson's sayings was that if you find an integer in operator theory you should look for a Fredholm operator. Arveson partially extended Kadison's work on the Pythagorean Theorem in [3] where he studied the diagonals of normal operators with finite spectrum with infinite multiplicity that forms the vertices of a convex polygon in \(\mathbb{C}\text{,}\) infinite co-infinite projections being a degenerate special case. He also found an “index obstruction” for their diagonals which depended on the following result.

Here we use the notation \(\projone{} \wedge \projtwo{}\) to denote the largest projection less than both \(\projone{}\) and \(\projtwo{}\text{,}\) i.e., the projection onto \(\projone{}\h \cap \projtwo{}\h\text{.}\) Similarly, \(\projone{} \vee \projtwo{}\) denotes the smallest projection greater than both \(\projone{}\) and \(\projtwo{}\text{,}\) i.e., the projection onto the closure of \(\projone{}\h + \projtwo{}\h\text{.}\)

Whenever we have two projections, \(\projone{}\) and \(\projtwo{}\text{,}\) we denote by \(\projtwo{}\mid_{\projone{}\h}\) the operator in \(\B(\projone{}\h, \projtwo{}\h)\text{.}\) Then a key step in Arveson's proof is the fact that if \(\projone{}-\projtwo{}\in \mathcal L^2\text{,}\) then

Although Arveson did not state so explicitly, embedded in his proofs one can also find the fact that using the notations established above, if \(a+b\lt \infty\text{,}\) then indeed \(\projone{}-\projtwo{}\in \mathcal L^2\) and hence \(a-b= \idx(\projtwo{}\mid_{\projone{}\h})\text{,}\) which explains why \(a-b\) is an integer. What remains to be explained is the role of \(\projtwo{}\mid_{\projone{}\h}\) and the significance of its index. Note that \(\idx (\projtwo{}\mid_{\projone{}\h}) = -\idx (\projone{}\mid_{\projtwo{}\h})\) since \((\projtwo{}\mid_{\projone{}\h})^{*} = \projone{}\mid_{\projtwo{}\h}\text{.}\)

A similar question arises from another proof that \(a-b\) is an integer which was obtained in [19]. Let us briefly sketch the original computation (reformulated in new notation) as it introduces the connections we want to illustrate.

Let \(w\) be an isometry with range \(\projone{}\text{,}\) let \(\Lambda:=\{j\mid d_j> 1/2 \}\text{,}\)

Then \(\|f_j\|=1\) for all \(j\text{,}\) \(t_a, t_b\in \mathcal L^1_+\text{,}\) and \(1 = \sum_j d_j f_j \otimes f_j\text{,}\) hence

where \(E\) denotes the conditional expectation on the algebra of diagonal operators with respect to the orthonormal basis \(\{e_j\}\text{,}\) namely \(E(x)\) is the diagonal of an operator \(x\in\B(\h)\text{.}\) Hence

It is then immediate to see (but was not remarked explicitly in [19]), that

Notice that \(f\) can be interpreted as the analysis operator of the Bessel sequence \(\{f_j\}_{j\in \Lambda}\) and \(v\) as the analysis of the canonical Parseval frame. While this construction provides indeed a proof that \(v^*\mid_{\projtwo{}\h}\) is Fredholm, a natural question is why \(\idx (v^*\mid_{\projtwo{}\h})= \idx (\projone{}\mid_{\projtwo{}\h})\) as can be obtained from Arveson's work. To answer it, notice first that since \(f^*f\) is a trace-class perturbation of the identity, it is Fredholm and hence so are \(|f|\) and \(f^*=|f|v^*\text{.}\) Furthermore \(\idx (f^*\mid{\projtwo{}\h})= \idx(v^*\mid{\projtwo{}\h})\) since \(\idx(|f|)=0\text{.}\) Next

where \(d:= \sum _{j\in \Lambda}\sqrt{d_j}e_j\otimes e_j\ge \frac{1}{\sqrt{2}} \projtwo{}\) is invertible in \(\B(\projtwo{}\h)\text{.}\) Thus \(w^*\mid_{\projtwo{}\h}\) is also Fredholm in \(\B(\projtwo{}\h, \h)\) and

It is then immediate to verify (see also ((2.4)) below) that \(\idx (w^*\mid_{\projtwo{}\h})= \idx (\projone{}\mid_{\projtwo{}\h})\) as obtained by Arveson.

However, neither the proof due to Arveson nor the one in [19]) provides a natural explanation of the role of \(w^*\mid_{\projtwo{}\h}\) or \(\projone{}\mid_{\projtwo{}\h}.\)

The goal of our paper is to provide an explanation of that role in the context of the notion of essential codimension \([\projone{}:\projtwo{}]\) of a pair of projections \(\projone{}\) and \(\projtwo{}\) with \(\projone{}-\projtwo{}\in \K\) that was introduced in the BDF theory (see [8] and Section 2), or of the more general notion of index of a Fredholm pair of projections, introduced by Avron, Seiler, and Simon in [4].

Combining Arveson's work with the study of Fredholm pairs and essential codimension, one can provide a natural identification of Kadison's integer with the essential codimension of a pair of projections. In the notations of Theorem 1.1 we have:

To understand the simple proof of this result, and for the convenience of the readers not familiar with the notions of Fredholm pairs, essential codimension, and the work of Arveson in [3], we will provide in Section 2 a self-contained short presentation of the relevant results of the theory of Fredholm pairs. We have strengthened several results and generalized them to the case when \(\projone{}-\projtwo{}\) belongs to an arbitrary (two-sided) operator ideal \(\mathcal J\) rather than just the Hilbert–Schmidt ideal \(\mathcal L^2\text{.}\)

Since Fredholm pairs have found most of their applications in the theory of spectral flows in type I or type II von Neumann algebras, we will conclude Section 2 with a very brief foray into the case when the notion of Fredholm operators and indices are taken relative to a semifinite von Neumann algebra (also called Breuer–Fredholm, or more precisely \(\tau\)-Breuer–Fredholm operators).

In Section 3 we will assemble the results previously collected into a proof of Theorem 1.3 that is inspired by, but independent of, the work by Arveson. Then we will extend part of Proposition 2.8 to positive contractions. We will then use the same techniques to identify an integer appearing in the study by Bownik and Jasper of the diagonals of selfadjoint operators with finite spectrum and also to simplify the proof of one of the key results of that paper (see [6]).

We thank R. Douglas for having suggested to the first named author of this paper to consider a possible connection between the frame approach originally used and essential codimension.