New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Diagonals of normal operators with finite spectrum

Edited by I. M. Singer, Massachusetts Institute of Technology, Cambridge, MA, and approved November 8, 2006 (received for review June 27, 2006)
Abstract
Let X = {λ_{1}, …, λ_{N}} be a finite set of complex numbers, and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e_{1}, e_{2}, … for H, A gives rise to a matrix whose diagonal is a sequence d = (d_{1}, d_{2}, …) with the property that each of its terms d_{n} belongs to the convex hull of X. Not all sequences with that property can arise as the diagonal of a normal operator with spectrum X. The case where X is a set of real numbers has received a great deal of attention over the years and is reasonably well (though incompletely) understood. In this work we take up the case in which X is the set of vertices of a convex polygon in ℂ. The critical sequences d turn out to be those that accumulate rapidly in X in the sense that We show that there is an abelian group Γ _{X} , a quotient of ℝ^{2} by a countable subgroup with concrete arithmetic properties, and a surjective mapping of such sequences d ↦ s(d) ∈ Γ _{X} with the following property: If s(d) ≠ 0, then d is not the diagonal of any such operator A. We also show that while this is the only obstruction when N = 2, there are other (as yet unknown) obstructions when N = 3.
Given a selfadjoint n × n matrix A, the diagonal of A and the eigenvalue list of A are two points of ℝ^{n} that bear some relation to each other. The Schur–Horn theorem characterizes that relation in terms of a system of linear inequalities (1, 2). That characterization has attracted a great deal of interest over the years and has been generalized in remarkable ways. For example, refs. 3–6 represent some of the milestones. More recently, a characterization of the diagonals of projections acting on infinite dimensional Hilbert spaces has been discovered (7, 8), and a version of the Schur–Horn theorem for positive traceclass operators is given in ref. 9. The latter reference contains a somewhat more complete historical discussion.
Let X be a finite subset of the complex plane ℂ, and consider the set (X) of all normal operators acting on a separable Hilbert space H that have spectrum X with uniformly infinite multiplicity, The set (X) is invariant under the action of the group of *automorphisms of (H), and it is closed in the operator norm. Fixing an orthonormal basis e _{1}, e _{2}, … for H, one may consider the (nonclosed) set (X) of all diagonals of operators in (X) In this paper we address the problem of determining the elements of (X).
Notice that for every sequence d = (d _{1}, d _{2}, …) in (X), each term d_{n} must belong to the convex hull of X. Indeed, since there is a normal operator A with spectrum X such that d_{n} = 〈Ae_{n} , e_{n} 〉, n ≥ 1, each d_{n} must belong to the numerical range of A, and the closure of the numerical range of a normal operator is the convex hull of its spectrum.
This necessary condition d_{n} ∈ conv X, n ≥ 1, is not sufficient. Indeed, a characterization of ({0, 1}) (the set of diagonals of projections) was given by Kadison (8), the main assertion of which can be paraphrased as follows:
Theorem 1 (Theorem 15 of ref. 8).
Let d = (d _{1}, d _{2}, …) ∈ ℓ^{∞} be a sequence satisfying 0 ≤ d_{n} ≤ 1 for every n and Let a, b ∈ [0, ∞] be the numbers Then one has the following dichotomy:

If a + b = ∞, then d ∈ ({0, 1}).

If a + b < ∞, then d ∈ ({0, 1}) ⇔ a − b ∈ ℤ.
In a recent paper (9), a related spectral characterization was found for the possible diagonals of positive traceclass operators. That paper did not address the case of more general selfadjoint operators, and in particular, the results of ref. 9 shed no light on the phenomenon (ii) of Theorem 1. This paper grew out of an effort to understand that phenomenon as an index obstruction. We achieve that for certain finite subsets X ⊆ ℂ in place of {0, 1}, namely those that are affinely independent in the sense that none of the points of X can be written as a nontrivial convex combination of the others, i.e., when X is the set of vertices of a convex polygon.
The basic issues taken up here bear some relation to A. Neumann's work (10) on the infinitedimensional Schur–Horn theorem for selfadjoint operators. But there is a fundamental difference in the nature of the characterizations of ref. 10 and the results below that goes beyond the fact that Neumann confines attention to selfadjoint operators. The comparison is clearly seen for the twopoint set X = {0, 1}. In that case, the results of ref. 10 provide the following description of the closure of (X) in the ℓ^{∞}norm: (see lemma 2.13 and proposition 3.12 of ref. 10). Thus, the exceptional cases described in part (ii) of Theorem 1 disappear when one passes from (X) to its closure in the ℓ^{∞}norm. In more explicit terms, while sequences d = (d_{n} ) satisfying 0 ≤ d_{n} ≤ 1, n ≥ 1, a + b < ∞, a − b ∉ ℤ, fail to belong to (X), they are all absorbed into its normclosure.
It is these “exceptional” cases that we seek to understand here, for more general finite sets X ⊆ ℂ. Our main result (Theorem 4 below) identifies an index obstruction corresponding to (ii) above when X is the set of vertices of a convex polygon P. Specifically, consider the set of all sequences d = (d_{n} ) that satisfy d_{n} ∈ P, n = 1, 2, …, and which accumulate rapidly in X in the precise sense that We show that there is a discrete abelian group Γ _{X} , depending only on the arithmetic properties of X, and a surjective mapping d ↦ s(d) ∈ Γ _{X} of the set of all such sequences d, with the following property: If s(d) ≠ 0, then d is not the diagonal of any operator in (X). We use Theorem 1 to show that this is the only obstruction in the case of two point sets, but we also show by example that there are other (as yet unknown) obstructions in the case of threepoint sets.
Sequences in Lim^{1}(X) and the Group Γ _{X}
Let X = {λ_{1}, …, λ_{N}} be a finite set of complex numbers. For every complex number z we write for the distance from z to X. We consider the space Lim^{1}(X) of all sequences a = (a _{1}, a _{2}, …) ∈ ℓ^{∞} with the property Thus, a sequence a = (a_{n} ) belongs to Lim^{1}(X) iff all of its limit points belong to X and it converges rapidly to its limit points in the following sense: there is a sequence x = (x_{n} ) satisfying x_{n} ∈ X for every n = 1, 2, …, and In the context of Theorem 1, Eq. 2 reduces to the hypothesis of (ii) when X = {0, 1} (see Concluding Remarks and an Example). In this section we show that every element a ∈ Lim^{1}(X) has a “renormalized” sum that takes values in an abelian group Γ_{X} naturally associated with X.
For fixed a ∈ Lim^{1}(X) there are many Xvalued sequences x = (x_{n} ) that satisfy Eq. 2 . Nevertheless, one can attempt to define a renormalized sum of an element a ∈ Lim^{1}(X) by choosing a sequence x_{n} ∈ X that satisfies Eq. 2 and forming the complex number While the value of s depends on the choice of x ∈ X, the following observation shows that the ambiguity is associated with a countable subgroup of the additive group of ℂ.
Proposition 1.
Let X = {λ_{1}, …, λ_{N}} be a finite subset of ℂ and fix a = (a_{n} ) ∈ Lim ^{1}(X). For any two sequences x = (x_{n} ), y = (y_{n} ) of points in X that satisfy the sequence of differences x − y = (x_{n} − y_{n} ) is finitely nonzero, and there are integers ν_{1}, …, ν_{N} ∈ ℤ such that ν_{1} + ν_{2} + ⋯ + ν_{N} = 0 and
Proof.
Since a − y and a − x both belong to ℓ^{1}, their difference x − y must also belong to ℓ^{1}. Since x_{n} − y_{n} takes values in the finite set of differences X − X and belongs to ℓ^{1}, it must vanish for all but finitely many n, and for each of the remaining n, x_{n} − y_{n} is of the form λ _{i} _{n} − λ _{j} _{n} where i_{n} , j_{n} ∈ {1, 2, …, N}. It follows that is a finite sum of terms of the form λ _{i} − λ _{j} , 1 ≤ i, j ≤ N, and such a number has the form 3 with integer coefficients ν_{k} ∈ ℤ having sum 0.
Definition 1 (The Obstruction Group Γ_{X}).
For every finite set of N ≥ 2 complex numbers X = {λ_{1}, …, λ_{N}}, let K_{X} be the additive subgroup of ℂ consisting of all z of the form where ν_{1}, …, ν_{N} ∈ ℤ satisfy ν_{1} + ⋯ + ν_{N} = 0. Γ_{X} will denote the quotient of abelian groups
K_{X} is the subgroup of ℂ generated by the set of differences λ_{i} − λ_{j}, for i, j = 1, …, N, or equivalently by {λ_{2} − λ_{1}, λ_{3} − λ_{1}, …, λ_{N} − λ_{1}}. Hence, the rank of K_{X} is at most N − 1. The rank is N − 1 iff when one views ℂ as a vector space over the field ℚ of rational real numbers, the set of differences {λ_{2} − λ_{1}, λ_{3} − λ_{1}, …, λ_{N} − λ_{1}} becomes a linearly independent set.
By Proposition 1, we can define a map s : Lim^{1}(X) → Γ_{X} as follows: For every a = (a_{n} ) ∈ Lim^{1}(X), choose a sequence x = (x_{n} ) that takes values in X and satisfies Σ_{n} a_{n} − x_{n}  < ∞, and let s(a) be the coset
Definition 2 (Renormalized Sum).
For every sequence a ∈ Lim^{1}(X), the element s(a) ∈ Γ_{X} is called the renormalized sum of a.
When it is necessary to call attention to the set X of vertices, we will write s_{X} (a) rather than s(a).
Remark 1 (surjectivity of the map s: Lim^{1}(X) → Γ_{X}).
One thinks of Γ_{X} as an uncountable discrete abelian group. It is easy to see that the map s is surjective. Indeed, for any z ∈ ℂ the coset z + K_{X} ∈ Γ_{X} is realized as the value s(a) of a renormalized sum as follows. Choose any sequence u = (u_{n} ) in ℓ^{1} such that and let x = (x_{n} ) be an arbitrary sequence satisfying x_{n} ∈ X for every n ≥ 1. Then the sum a = u + x belongs to Lim^{1}(X) and satisfies s(a) = z + K_{X} .
We require the following elementary description of sequences in Lim^{1}(X).
Proposition 2.
Let X = {λ_{1}, …, λ_{N}} be a finite set of complex numbers and let a = (a_{n} ) ∈ ℓ^{∞}. Then the following are equivalent:

a ∈ Lim^{1}(X).

One has the summability condition
where f is the polynomial f(z) = (z − λ_{1})(z − λ_{2}) ⋯ (z − λ_{N}).
Proof.
Let be the minimum distance between distinct points of X. Note first that whenever z ∈ ℂ satisfies d(z, X) ≤ δ/2, one has Indeed, if we choose k such that d(z, X) = z − λ_{k}, then for j ≠ k we have hence, and Eq. 4 follows.
If a = (a _{1}, a _{2}, …) is a sequence such that Σ_{n}f(a_{n} ) converges, then since f(z) ≥ d(z, X)^{N} for all z ∈ ℂ, it follows that d(a_{n} , X)^{N} → 0 as n → ∞, hence, there is an n _{0} such that d(a_{n} , X) ≤ δ/2 for n ≥ n _{0}. Eq. 4 implies so that Σ_{n} d(a_{n} , X) converges, hence, a ∈ Lim^{1}(X).
Conversely, assuming that Σ_{n} d(a_{n} , X) converges, let Since d(a_{n} , X) → 0, we can find n _{1} such that a_{n} ≤ 2R for n ≥ n _{1}. Choosing k _{1}, k _{2}, … so that d(a_{n} , X) = a_{n} − λ _{k} _{n}  we have for all n ≥ n _{1}, hence Σ_{n}f(a_{n} ) converges.
XDecompositions
We view ℓ^{∞} as a commutative C*algebra with unit 1, and the elements of ℓ^{∞} as bounded functions a : ℕ → ℂ, with norm Let P be a convex polygon in the complex plane and let X = {λ_{1}, …, λ_{N}} be its set of vertices, with N ≥ 2. In this section, we describe an elementary (nonunique) decomposition for sequences that take values in P, and we show that under certain circumstances, all such decompositions of a sequence must share a key property.
Proposition 3.
Let P ⊆ ℂ be a convex polygon with vertices {λ_{1}, …, λ_{N}}. Every sequence a ∈ ℓ^{∞} satisfying a(n) ∈ P, n ≥ 1, can be decomposed into a sum of the form where e _{1}, …, e_{N} are positive elements of ℓ^{∞} satisfying e _{1} + ⋯ + e_{N} = 1.
Conversely, any sequence a of the form 5 , with positive elements e_{k} summing to 1, must satisfy a(n) ∈ P for every n ≥ 1.
Proof.
Fix a and choose n ≥ 1. Since a(n) ∈ P and P is the convex hull of {λ_{1}, …, λ_{N}}, we can find a point (e _{1}(n), …, e_{N} (n)) ∈ ℝ^{N} such that e_{1} (n) ≥ 0, …, e_{N} (n) ≥ 0, e _{1}(n) + ⋯ + e_{N} (n) = 1, and The sequences e_{k} = (e_{k} (1), e_{k} (2), …) ∈ ℓ^{∞} satisfy e_{k} ≥ 0, e_{1} + ⋯ + e_{N} = 1, and the asserted representation 5 follows.
The converse assertion is obvious.
Definition 3.
Let P ⊆ ℂ be a convex polygon whose set of vertices is X = {λ_{1}, …, λ_{N}}, and let a ∈ ℓ^{∞} satisfy a(n) ∈ P, n ≥ 1. A representation of the form 5 is called an Xdecomposition of a.
Despite the fact that Xdecompositions are not unique except in very special circumstances, there is a property common to all Xdecompositions of a in cases where That result (Theorem 2 below) requires the case n = 2 of the following.
Lemma 1.
Let P be a convex polyhedron in ℝ^{n} with extreme points x _{1}, …, x_{r} . Consider the simplex and the affine map of Δ onto P defined by For any choice of norms on ℝ^{r} and ℝ^{n}, there is a constant C > 0 such that where d(v, S) = inf{‖v − s‖ : s ∈ S} denotes the distance from a vector v to a set S, and where δ_{1}, …, δ_{r} are the extreme points of Δ, (δ_{k})_{j} = δ_{kj}.
Proof.
It suffices to show that for each k = 1, …, r, there is a constant C_{k} such that Indeed, Eq. 7 implies where C = max(C _{1}, …, C_{r} ), and Eq. 6 follows.
By symmetry, it suffices to prove Eq. 7 for k = 1; moreover, after performing an affine translation if necessary, there is no loss of generality if we assume that x _{1} = 0 is one of the extreme points of P. For every extreme point e of a convex polyhedron P, there is a supporting hyperplane that meets P only at {e}. Thus, there is a linear functional f on ℝ^{n} such that f(x) > 0 for all nonzero x ∈ P. For each t ∈ Δ we have After noting that f(x(t)) ≤ ‖f‖ · ‖x(t)‖, we obtain and Eq. 7 follows.
Theorem 2.
Let X = {λ_{1}, …, λ_{N}} be the set of vertices of a convex polygon P ⊆ ℂ and let a ∈ ℓ^{∞} satisfy a(n) ∈ P for n ≥ 1. If a ∈ Lim^{1}(X), then for every Xdecomposition of the form 5 each of the sequences e _{1}, …, e_{N} belongs to Lim^{1}{0, 1}.
Proof.
Consider the Euclidean norm on ℂ, the norm on ℝ^{N}, and fix n = 1, 2, …. Since the point (e _{1}(n), …, e_{N} (n)) ∈ ℝ^{N} belongs to the simplex Δ of Lemma 1, there is a constant C > 0 such that Since for every point t = (t _{1}, …, t_{N} ) in the simplex Δ and for every fixed k = 1, …, N we have it follows that Using a ∈ Lim^{1}(X), we can sum the preceding inequality on n to obtain Hence, e_{k} ∈ Lim^{1}{0, 1}.
Two Projections
It is known that for any pair of projections P, Q ∈ (H) for which P − Q is a traceclass operator, trace(P − Q) must be an integer. For example, the result can be found in Effros' article (see lemma 4.1 in ref. 11). We require an appropriate extension of that result to the case where P − Q is merely a Hilbert–Schmidt operator, Theorem 3 below. Throughout the remainder of this paper, we write ^{1} (resp. ^{2}) for the Banach space of traceclass operators (resp. Hilbert–Schmidt operators) acting on a given Hilbert space, and we write P ^{⊥} for 1 − P when P ∈ (H) is a projection.
Theorem 3.
Let M, N be subspaces of a Hilbert space H with respective projections P, Q, and assume that P − Q ∈ ^{2}.
Then both Q(P − Q)Q and Q ^{⊥}(P − Q)Q ^{⊥} belong to ^{1}, both subspaces M ∩ N ^{⊥} and N ∩ M^{⊥} are finitedimensional, and In particular, QPQ + Q ^{⊥} PQ ^{⊥} − Q is a traceclass operator such that
In the proof we will show that the left side of Eq. 8 is the index of a Fredholm operator, and for that we require the following elementary result for which we lack a convenient reference:
Lemma 2.
Let H, K be Hilbert spaces and let A : H → K be an operator such that both 1 _{H} − A* A and 1 _{K} − AA* are traceclass. Then A is a Fredholm operator in (H, K) whose index is given by the formula
Proof.
Consider the polar decomposition A = UB, where B is a positive operator and U is a partial isometry with initial space ker A ^{⊥} and range r̅a̅n̅ ̅A̅ = ker A*^{⊥}. Let C be the restriction of B ^{2} to ker A ^{⊥}. Then where C′ is unitarily equivalent to C; indeed, the restriction of U to ker A ^{⊥} implements a unitary equivalence of C and C′.
It follows that 1 − A* A = (1 − C) ⊕ 1 _{ker A}, and 1 − AA* is unitarily equivalent to (1 − C) ⊕ 1 _{ker A*}. Since 1 − C is a traceclass operator, we have trace (1 − A* A) = trace(1 − C) + dim ker A, and similarly trace (1 − AA*) = trace(1 − C) + dim ker A*. The terms involving trace(1 − C) cancel, and as asserted.
Proof of Theorem 3.
We claim first that Q(P − Q)Q ∈ ^{1}. Indeed, we have Q(P − Q)Q = −(Q − QPQ), and Q − QPQ is a positive operator satisfying Similarly, Q ^{⊥}(P − Q)Q ^{⊥} = Q ^{⊥} PQ ^{⊥}, and so that Q ^{⊥}(P − Q)Q ^{⊥} ∈ ^{1}.
Let H _{0} be the subspace of H spanned by the mutually orthogonal subspaces M ∩ N ^{⊥} and N ∩ M ^{⊥}. The restriction of P − Q to H _{0} is unitary with eigenvalues ±1; hence, dim H _{0} = trace(P − Q)P_{H} _{0}^{2} ≤ traceP − Q^{2} must be finite.
Consider the operator A : N → M defined by restricting P to N = QH. Obviously, ker A = N ∩ M ^{⊥} and ker A* = M ∩ N ^{⊥}, and we claim that A satisfies the hypotheses of Lemma 2. Indeed, Since P − Q is Hilbert–Schmidt, Q ^{⊥} P = (P − Q)P and P ^{⊥} Q = (Q − P)Q are both Hilbert–Schmidt; hence, Q ^{⊥} P^{2} and P ^{⊥} Q^{2} are both traceclass. Thus, we can apply Eq. 10 to obtain The left side is dim ker A − dim ker A* = dim(N ∩ M ^{⊥}) − dim(M ∩ N ^{⊥}), and Eq. 8 follows from the preceding formula.
Projections with Diagonals in Lim^{1}{0, 1}
In this section we characterize the projections in (H) whose diagonals relative to a given orthonormal basis belong to Lim^{1}{0, 1}.
Proposition 4.
Let e _{1}, e _{2}, … be an orthonormal basis for a Hilbert space H, let be the maximal abelian von Neumann algebra of all operators that are diagonalized by (e_{n} ), and let E : (H) → be the tracepreserving conditional expectation For every projection P ∈ (H), the following are equivalent:

The diagonal of P relative to the basis (e_{n} ) belongs to Lim^{1}{0, 1}.

E(P) − E(P)^{2} ∈ ^{1}.

P ∈ + ^{2}.
The proof of Proposition 4 requires the following formula:
Lemma 3.
Let E : (H) → be the map (Eq. 11). Then for every projection P ∈ (H) we have
Proof of Lemma 3.
We have E(P)e_{n} = d_{n}e_{n} , where d_{n} = 〈Pe_{n} , e_{n} 〉. Since d_{n}e_{n} is the projection of Pe_{n} onto the onedimensional space ℂ · e_{n} , we have Hence, and the right side is evidently the trace of E(P) − E(P)^{2}.
Proof of Proposition 4.
Let d = (d _{1}, d _{2}, …) be the diagonal of P relative to (e_{n} ), d_{n} = 〈Pe_{n} , e_{n} 〉, n ≥ 1. Then hence the equivalence of (i) and (ii) follows from Proposition 2.
(iii) ⇒ (ii): Assume first that the projection P can be decomposed into a sum P = A + T where A ∈ and T is Hilbert–Schmidt. Then P − E(P) = T − E(T), and T − E(T) is a Hilbert–Schmidt operator. By Eq. 12 , we obtain
(ii) ⇒ (iii): Assume that trace(E(P) − E(P)^{2})) < ∞, and consider the operator T = P − E(P). By Eq. 12 , we have so that T is Hilbert–Schmidt. Thus, P = E(P) + T ∈ + ^{2}.
Diagonals of Operators in (X)
We are now in position to prove our main result. Let X = {λ_{1}, …, λ_{N}} be the set of vertices of a convex polygon P ⊆ ℂ, and let (X) be the set of all normal operators A acting on a separable Hilbert space H that have spectrum X with infinite multiplicity Fix an orthonormal basis e _{1}, e _{2}, … for H. There are two necessary conditions that a sequence d = (d _{1}, d _{2}, …) ∈ ℓ^{∞} must satisfy for it to be the diagonal of an operator A ∈ (X), d_{n} = 〈Ae_{n} , e_{n} 〉, n ≥ 1, namely:

d_{n} ∈ P for every n ≥ 1,

d has an Xdecomposition
in which Σ_{n=1} ^{∞} E_{k} (n) = ∞ for every k = 1, …, N. Indeed the projections P_{k} arising from the spectral representation of A have diagonals E_{k} (n) = 〈P_{k}e_{n} , e_{n} 〉 that give rise to an Xdecomposition with the property (ii). The requirements (i), (ii) on a sequence do not guarantee that it is the diagonal of an operator in (X). We now identify an obstruction that emerges when d ∈ Lim^{1}(X) and which involves the renormalized sum s : Lim^{1}(X) → Γ_{X} of Definition 1.
Theorem 4.
Let X = {λ_{1}, …, λ_{N}} be the set of vertices of a convex polygon P ⊆ ℂ and let d = (d _{1}, d _{2}, …) be a sequence of complex numbers satisfying d_{n} ∈ P, n ≥ 1, together with the summability condition where f(z) = (z − λ_{1})(z − λ_{2}) ⋯ (z − λ_{N}). Then d ∈ Lim^{1}(X); and if d is the diagonal of an operator in (X), then s(d) = 0.
Proof.
By Proposition 2, the summability condition 13 characterizes sequences in Lim^{1}(X).
Fix an orthonormal basis e _{1}, e _{2}, … for a Hilbert space H, and assume that there is an operator A ∈ (X) such that d_{n} = 〈Ae_{n} , e_{n} 〉, n = 1, 2, …. To show that s(d) = 0, we must find a sequence (b_{n} ) that takes values in X, satisfies Σ_{n} d_{n} − b_{n}  < ∞, and we must exhibit integers ν_{1}, …, ν_{n} satisfying ν_{1} + ⋯ + ν_{N} = 0, and For that, consider the maximal abelian algebra of all operators that are diagonalized by the basis e _{1}, e _{2}, …, let E : (H) → be the tracepreserving conditional expectation and let D = E(A) ∈ be the operator We must find an operator B ∈ ∩(X) such that D − B ∈ ^{1}, and integers ν_{1}, …, ν_{N} summing to zero, such that
The latter are achieved as follows. Let be the spectral representation of A, with P _{1}, …, P_{N} a set of mutually orthogonal infinite rank projections with sum 1. Then we have so that E(P _{1}), …, E(P_{N} ) define an Xdecomposition of D. Theorem 2 implies that when one views the operators E(P_{k} ) as sequences in ℓ^{∞}, one has E(P _{k}) ∈ Lim^{1}({0, 1}) for each k = 1, …, N.
We claim that there is a sequence Q _{1}, …, Q_{N} of mutually orthogonal projections in having sum 1, which satisfy Indeed, since E(P_{k} ) ∈ Lim^{1}({0, 1}) for each k, the definition of Lim^{1}({0, 1}) implies that we can find projections Q _{1} ^{0}, …, Q _{N} ^{0} ∈ such that Considering each Q _{k} ^{0} as a sequence in ℓ^{∞} that takes values in {0, 1}, the sum Q _{1} ^{0} + ⋯ + Q _{N} ^{0} is a sequence taking values in {0, 1, 2, …, N}. Consider the set S = {n ∈ ℕ : Σ_{k=1} ^{N} Q _{k} ^{0}(n) = 1} ⊆ ℕ. Since P _{1} + ⋯ + P_{N} = 1 we have E(P _{1}) + ⋯ + E(P_{N} ) = 1, and hence It follows that The latter implies that ℕ\ S is a finite set, and that Q _{1} ^{0} · χ_{S}, …, Q _{N} ^{0} · χ_{S} are mutually orthogonal projections with sum χ_{S}. Thus, if we modify the sequence Q _{1} ^{0}, …, Q _{N} ^{0} as follows, we obtain a new sequence of projections Q _{1}, …, Q_{N} ∈ which are mutually orthogonal, have sum 1, and satisfy Eq. 15 .
Note too that since trace P_{k} = rank P_{k} = ∞ for every k, Eq. 15 implies that trace Q_{k} = rank Q_{k} = ∞ as well. It follows that the operator belongs to , satisfies σ(B) = σ_{e}(B) = X, and by construction,
It remains to show that trace(D − B) satisfies Eq. 14 for integers ν _{k} as described there. Indeed, since D − B = Σ _{k} λ _{k} (E(P_{k} ) − Q_{k} ) and E(P_{k} ) − Q_{k} belongs to ^{1}, we have so it suffices to show that and that the sum of the N integers of Eq. 17 is 0.
To prove Eq. 17 we appeal to Theorem 3. Note first that P − E(P) belongs to ^{2}. Indeed, since E(P_{k} ) − Q_{k} is traceclass and Q_{k} is a projection, we have trace (E(P) − E(P)^{2}) < ∞, so by Eq. 12 , Since E(P_{k} ) − Q_{k} ∈ ^{1} ⊆ ^{2}, we obtain From Theorem 3 we conclude that and moreover Since E( ^{1}) ⊆ ^{1} ∩ and since we find that E(P_{k} ) − Q_{k} ∈ ^{1} and Since Σ_{k=1} ^{N} (E(P_{k} ) − Q_{k} ) = E(1) − 1 = 0, we have ν_{1} + ⋯ + ν_{N} = 0. Finally, and Eq. 14 follows.
Concluding Remarks and an Example
We point out that Theorem 4 specializes to the assertion (ii) ⇒ of Theorem 1 in the case X = {0, 1}. Indeed, a straightforward calculation shows that for the twopoint set X = {0, 1} one has K
_{{0,1}} = ℤ, so that Γ_{{0,1}} = ℂ/ℤ = 𝕋 × ℝ. Now the hypothesis of Theorem 1 (ii) is that a + b < ∞, where a and b are defined by
Let (x_{n}
) be the sequence x_{n}
= 0 when d_{n}
≤
For more general sets X, the converse of Theorem 4 would assert:
Let X be the set of vertices of a convex polygon and let d be a sequence in Lim^{1}(X) such that s_{X} (d) = 0. Then there is an operator N ∈ (X) and an orthonormal basis e _{1}, e _{2}, … for H such that d_{n} = 〈Ne_{n} , e_{n} 〉, n ≥ 1.
We first point out that this converse is true when X consists of just two points. To sketch the argument briefly, suppose X = {λ_{1}, λ_{2}} with λ_{1} ≠ λ_{2}. One can find an affine bijection z ↦ az + b of ℂ that carries λ_{1} to 0 and λ_{2} to 1, and which therefore carries sequences in Lim^{1}(X) to sequences in Lim^{1}({0, 1}). After noting that the operator mapping T ↦ aT + b 1 carries (X) to ({0, 1}), one can make use of the implication ⇐ of Theorem 1 (ii) in a straightforward way to deduce the required result.
On the other hand, the following example shows that this converse of Theorem 4 fails for threepoint sets.
Proposition 5.
Let X = {0, 1, i}, i denoting
Before giving the proof, we recall that a doubly stochastic n × n matrix A = (a_{ij} ) is said to be orthostochastic if there is a unitary n × n matrix (u_{ij} ) such that a_{ij} = u_{ij} ^{2}, 1 ≤ i, j ≤ n. We will make use of the following known example of a doubly stochastic matrix that is not orthostochastic (2).
Lemma 4.
The 3 × 3 matrix is not orthostochastic.
Proof.
Indeed, if there were a unitary 3 × 3 matrix U = (u_{ij} ) such that u_{ij} ^{2} = a_{ij} for all ij, then U must have the form with complex entries satisfying a = b = c = d = e = f = 1. But the rows of such a matrix cannot be mutually orthogonal.
Proof of Proposition 5.
Straightforward computations (that we omit) show that for the set X = {0, 1, i}, the group K_{X} and the obstruction group Γ_{X} are given by Let x be the sequence Obviously x_{n} ∈ X for every n = 1, 2, …, d_{n} = x_{n} except for n = 1, 2, 3, and Hence, d ∈ Lim^{1}(X) and s_{X} (d) = 0.
Every operator N ∈ (X) has the form N = P + iQ, where P, Q are mutually orthogonal infinite rank projections such that 1 − (P + Q) has infinite rank. Assuming that there is such an operator N whose matrix relative to some orthonormal basis e _{1}, e _{2}, … has diagonal d = (d _{1}, d _{2}, …), we argue to a contradiction as follows. Let p, q ∈ ℓ^{∞} be the real and imaginary parts of the sequence d Since d is the diagonal of P + iQ, one may equate real and imaginary parts to obtain p_{n} = 〈Pe_{n} , e_{n} 〉 and q_{n} = 〈Qe_{n} , e_{n} 〉, n = 1, 2, ….
Now for n ≥ 4, both p_{n} and q_{n} are {0, 1}valued. Since P and Q are projections, it follows that and in particular, both P and Q leave the closed linear span [e _{4}, e _{5}, e _{6}, … ] invariant. Hence, they leave its orthocomplement [e _{1}, e _{2}, e _{3}] invariant as well. Let P _{0}, Q _{0} be the restrictions of P, Q, respectively, to H _{0} = [e _{1}, e _{2}, e _{3}]. P _{0} and Q _{0} are mutually orthogonal projections, and the diagonals of their matrices relative to the orthonormal basis e _{1}, e _{2}, e _{3} are, respectively, Each of these two diagonals has sum 1; hence, P _{0} and Q _{0} are onedimensional. Moreover, R _{0} = 1 _{H0} − (P _{0} + Q_{0}) is a onedimensional projection in (H _{0}) whose diagonal relative to the basis e _{1}, e _{2}, e _{3} is Hence, the 3 × 3 matrix whose rows are the diagonals of the three projections P _{0}, Q _{0}, R _{0} takes the form If we now choose unit vectors f _{1}, f _{2}, f _{3} so that P _{0} = [f _{1}], Q _{0} = [f _{2}] and R _{0} = [f _{3}], we find that f _{1}, f _{2}, f _{3} is a second orthonormal basis for H _{0}, and where (u_{ij} ) is a unitary 3 × 3 matrix. This contradicts Lemma 4.
Proposition 5 shows that the necessary condition s_{X} (d) = 0 is not sufficient for a sequence d ∈ Lim^{1}(X) to be the diagonal of an operator in (X) when X contains more than two points. Moreover, the precise nature of the remaining obstructions when X consists of three noncolinear points remains mysterious.
Acknowledgments
I thank Richard Kadison, whose work (refs. 7 and 8) initially inspired this effort and with whom I have had the pleasure of many helpful conversations.
Footnotes
 ^{†}Email: arveson{at}math.berkeley.edu

Author contributions: W.A. wrote the paper.

The author declares no conflict of interest.

This article is a PNAS direct submission.
 © 2007 by The National Academy of Sciences of the USA
References

↵
 Schur I
 ↵

↵
 Kostant B

↵
 Atiyah M
 ↵
 ↵

↵
 Kadison R

↵
 Kadison R

↵
 Arveson W ,
 Kadison R
 Larson DR ,
 Han D ,
 Jorgensen PET
 ↵

↵
 Effros EG