Systems of two subspaces in a Hilbert space
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Tusi Mathematical Research Group
ORIGINAL PAPER
Systems of two subspaces in a Hilbert space Masatoshi Enomoto1 • Yasuo Watatani2 Received: 11 August 2020 / Accepted: 22 September 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract We study two subspace systems in a separable infinite-dimensional Hilbert space up to (bounded) isomorphism. One of the main result of this paper is the following: Isomorphism classes of two subspace systems given by graphs of bounded operators are determined by unitarily equivalent classes of the operator ranges and the nullity of the original bounded operators giving graphs. We construct several non-isomorphic examples of two subspace systems in an infinite-dimensional Hilbert space. Even if we study n subspace systems for n 3, we can use the analysis of any two subspaces of the n subspaces. Keywords Subspace Hilbert space Schatten class operator
Mathematics Subject Classification 46C07 47A15 16G20 16G60
1 Introduction Let E1 and E2 be two closed subspaces in a Hilbert space H, then we say that ðH; E1 ; E2 Þ is a two subspace system in H or a system of two subspaces in H. Let ðL; F1 ; F2 Þ be another two subspace system in L. We say that ðH; E1 ; E2 Þ and ðL; F1 ; F2 Þ are unitarily (resp. boundedly, algebraically) isomorphic if there exists a unitary operator (resp. bounded invertible operator, invertible operator) V from H to L such that VðE1 Þ ¼ F1 and VðE2 Þ ¼ F2 . Unitary isomorphism classes of two Communicated by Hiroyuki Osaka. & Yasuo Watatani [email protected] Masatoshi Enomoto [email protected] 1
Koshien University, Takarazuka, Hyogo 665-0006, Japan
2
Department of Mathematical Sciences, Kyushu University, Fukuoka, Japan
M. Enomoto and Y. Watatani
subspace systems are studied by many authors (cf. Araki [1], Davis [2], Dixmier [3], Halmos [8], Stone [14] etc.). It is easy to see that two subspace systems ðH; E1 ; E2 Þ and ðL; F1 ; F2 Þ are algebraically isomorphic if and only if HdimðE1 \ E2 Þ ¼ HdimðF1 \ F2 Þ, HdimðE1 =ðE1 \ E2 ÞÞ ¼ HdimðF1 =ðF1 \ F2 ÞÞ, HdimðE2 =ðE1 \ E2 ÞÞ ¼ HdimðF2 =ðF1 \ F2 ÞÞ and HdimðH=ðE1 þ E2 ÞÞ ¼ HdimðL=ðF1 þ F2 ÞÞ, where HdimðKÞ is a Hamel dimension of a vector space K. For a Hilbert space, we denote by dim H the Hilbert space dimension of H, that is, the cardinality of an orthonormal basis (or a complete orthonormal system) of H. These three types of isomorphisms (unitary isomorphisms, bounded isomorphisms and algebraic isomorphisms) are different each other. Unitary isomorphisms and bounded isomorphisms of two subspace systems are distinguished by angles. Bounded isomorphisms and algebraic isomorhisms of two subspace systems are also distinguished. For example, put an ¼ 1=n and bn ¼ 1=n2 . Let A be the diagonal operator with diagonals ðan Þn and B be the diagonal operator with diagonals ðbn Þn on K ¼ ‘2 ðNÞ. Put H ¼ K K: Then two subspace systems ðH; K 0; graphðAÞÞ and ðH; K 0; graphðBÞÞ are algebraically isomorphic, but not boundedly isomorphic, sinc
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