The norm attainment problem for functions of projections

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. This algebra contains every skew projection on that Hilbert space and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.

This note is in the spirit of paper [4]. The meta theorem of that paper is that the two projections theorem of Halmos is something like Robert Sheckley's Answerer: no question about the W * -and C * -algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. The norm attainment problem asks whether for a given bounded linear operator A there is a unit vector x such that Ax = A . In this generality, a useful answer is not available. Here we pose the question for the case where A is a function of two orthogonal projections or a function of one skew projection and its adjoint. Let P and Q be orthogonal projections acting on a real or complex Hilbert space H. According to Halmos' "Two projections theorem" (see [8] and consult [3,10] for the history and more on the subject), there is a representation of H as an orthogonal sum (the last two summands have the same dimension and are thus identified via an appropriate unitary similarity) with respect to which and Here and below we use the string (a 00 , a 01 , a 00 , a 11 ) as an abbreviation for a 00 I M 00 ⊕ a 01 I M 01 ⊕ a 10 I M 10 ⊕ a 11 I M 11 , while the blocks of the matrix component in (2) According to the Giles-Kummer theorem (see [7] or [3, Theorem 7.1]), the von Neumann algebra W * (P, Q) generated by P and Q consists of the operators A admitting the representation A = (a 00 , a 01 , a 00 , a 11 ) ⊕ φ 00 (H) φ 01 (H) with respect to the decomposition (1) of H. Here a ij are arbitrary complex numbers and φ ij are (also arbitrary) functions in L ∞ (σ(H)) with respect to the spectral measure of H. We will sometimes write the rightmost summand in (4) as Φ A (H). The norms of operators from W * (P, Q) were computed in [11,Theorem 10], see also [3,Theorem 7.9]. Namely, for A as in (4), Here The question we are addressing here is: when is A attained, i.e., when does there exist a unit vector x ∈ H such that Ax = A ? We will call A a norm attaining operator and write A ∈ N if this happens to be the case. For our purposes it is useful to recall that formula (5) was derived in [11] from the fact that is the right endpoint of the spectrum σ (Φ A * A (H)). Since for every operator X acting on H we have X ∈ N if and only if X * X ∈ N if and only if the right endpoint of σ(X * X) is its eigenvalue (see [6] and [9]), we just need to figure out when λ max is (or is not) an eigenvalue of A * A.
To this end, recall that for operators in W * (P, Q) the description of their kernels is also known ([3, Theorem 7.5] or [11,Theorem 1]). There is no need to include its exact form here, but an important for us consequence of it is as follows.
When applied to A − λI in place of A, Lemma 1 immediately yields the following.
Proposition 2 Let A be given by (4). Then λ ∈ C is an eigenvalue of A if and only if the spectral measure of the set {x ∈ σ(H) : Since trace Φ A * A = φ and det Φ A * A = |ω| 2 , the eigenvalues λ of A * A are characterized by the property that the spectral measure of the set {x ∈ σ(H) : In particular, λ max is an eigenvalue of A * A (and not just a point of its spectrum) if and only if the spectral measure of the set x ∈ σ(H) on which the function attains its maximum value is non-zero. Denoting this set by Σ(A), we arrive at the following conclusion.
Theorem 3 Let A be the operator given by (4). Then A ∈ N if and only if either (i) max (j,k)∈Λ |a jk | ≥ √ λ max or (ii) max (j,k)∈Λ |a jk | < √ λ max and the spectral measure of Σ(A) is non-zero.
Let now T be a skew projection on H. We assume that T is genuinely skew, which is equivalent to the requirement that T > 1. Denote by P = P Ran T the orthogonal projection onto the range of T and by Q = P Ker T the orthogonal projection onto the kernel of T . Afriat [1] (see also [3,Proposition 1.6]) showed that then P Q < 1 and Moreover, in (1)  We conclude with some examples. The authors of [2] recently proved that a skew projection T is in N if and only if the selfadjoint operator T + T * − I belongs to N . The operator T + T * − I appeared in [5] and is therefore called the Buckholtz operator in [2]. The following is an extension of this result.
Example 7 Let {ω n } ∞ n=1 be a sequence of positive real numbers that go monotonically to zero and let T be the skew projection on ℓ 2 (N) defined by the infinite matrix Then T ∈ N . Put A = T T * + T * T − T − T * − I. If ω n = 1/n, then A / ∈ N , but if ω n = 2/n, then A ∈ N .
Proof. It is clear that T ∈ N : the norm is attained at the vector . Similarly one can treat the operator A by sole calculations with 2 × 2 matrices. Here is how Theorem 3 works. We may write where x −1 n − 1 = ω n , that is, x n = 1/(1 + ω 2 n ). Straightforward computation gives Thus, ψ(x) = 2(2 − x −1 ) 2 . If ω n = 1/n, then This time ψ(x) assumes its maximum at x = 1/5, the value of maximum being λ max = ψ(1/5)/2 = 3.24. It follows that Σ(A) = {1/5}, and since 1/5 is an eigenvalue of H, we deduce from Theorem 3(ii) that A ∈ N . The last example can be elaborated to great extent. However, we leave it with Israel M. Gelfand: "Explain this to me on a simple example; the difficult example I will be able to do on my own." (http://www.israelmgelfand.com/edu_work.html)