Unitary operators with decomposable corners
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00091-w ORIGINAL PAPER
Unitary operators with decomposable corners Esteban Andruchow1,2 Received: 24 April 2020 / Accepted: 10 September 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract We study pairs (U, L0 ) , where U is a unitary operator in H and L0 ⊂ H is a closed subspace, such that
PL0 U|L0 ∶ L0 → L0 has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined. Keywords Unitary operator · Closed subspace · Singular value decomposition Mathematics Subject Classification 47AXX · 47A20 · 47B35
1 Introduction In this paper, we consider pairings (U, L0 ) of a unitary operator U in a Hilbert space H and a closed subspace L0 ⊂ H such that
PL0 U|L0 ∶ L0 → L0 admits a singular value decomposition (or shortly, is S-decomposable, meaning Schmidt decomposable). Here PL0 denotes the orthogonal projection onto L0 . Note that this condition is equivalent to say that PL0 UPL0 is S-decomposable. A typical
Communicated by Catalin Badea. * Esteban Andruchow [email protected] 1
Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, 1613 Los Polvorines, Argentina
2
Instituto Argentino de Matemática ‘Alberto P. Calderón‘, Saavedra 15, 3er. piso, 1083 Buenos Aires, Argentina
Vol.:(0123456789)
E. Andruchow
case of this situation occurs when L0 is an invariant subspace for U: in this case PL0 U|L0 = U|L0 is an isometry. There is a spatial characterization of this condition (see Corollary 2 below): PL0 U|L0 is S-decomposable if and only if there exist bi-orthonormal bases of L0 and UL0 , i.e., bases {fn ∶ n ≥ 1} of L0 and {gn ∶ n ≥ 1} of UL0 such that ⟨fn , gm ⟩ = 0 if n ≠ m. The problem is related to the characterization of pairs of projections P, Q such that PQ is S-decomposable, or equivalently, PQP is diagonalizable. Indeed, PL0 UPL0 is S-decomposable if and only if PL0 (UPL0 U ∗ ) is S - decomposable. We shall establish characterizations and abstract results concerning these pairings (U, L0 ): • Relations with the geometry of the Grassmann manifold of H : when does the expo-
nential map of the manifold eiZ L0 at a base point L0 give rise to a S-decomposable operator PL0 eiZ |L0 (Sect. 5). • Symmetries U (i.e. U ∗ = U −1 = U ) which have this property with respect to L0 . In particular, symmetries which arise from non-orthogonal projections (Sect. 6). • The relationship with diagonalizable dilations (Sect. 7). But also our interest will be in several concrete examples: • Multiplication by continuous unimodular functions in H = L2 (𝕋 ) and L0 = H 2 (𝕋 ). • H = L2 (ℝ) , U the Fourier-Plancherel transform and L0 = L2 (I) , where I is an
interval or the half line.
• H = 𝓁 2 (ℤ) and U = S the bilateral shift, L0 ⊂ 𝓁 2 (ℤ) a closed subspace.
The contents of the paper are the following: In Sect. 2, we recall preliminaries and establish basic properties. We denote the fact that PL0 U|L0 is
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