Strict convexity of the singular value sequences

If A and B are compact operators on a Hilbert space, with singular values satisfying s j (A) = s j (B) = s j ((A + B)/2) Let A be a compact linear operator from a Hilbert space H into a Hilbert space K. The singular values s 1 (A) ≥ s 2 (A) ≥. .. are the eigenvalues of |A| := (A * A) 1/2. We refer to [3] for other equivalent definitions and basic properties. In this note we offer two proofs, geometric and analytic, of the following uniqueness property of compact operators between Hilbert spaces. T h e o r e m. If A and B are compact operators such that s j (A) = s j (B) = s j (t 0 A + (1 − t 0)B) for some 0 < t 0 < 1 and all j = 1, 2,. .. , then A = B. σ k (A) := s 1 (A) + s 2 (A) +. .. + s k (A) are constant on the segment {tA + (1 − t)B : 0 ≤ t ≤ 1}. Therefore, each s j (·) is constant on this segment as well, for j = 1, 2,. . .. The analytic proof given below shows that the latter property extends to the whole real line connecting A and B. In particular, for j = 1, the norm tA + (1 − t)B is bounded for t → ∞, which is impossible unless A = B.