On the polar decomposition of an operator. (English) Zbl 0539.47014
An operator means a bounded linear operator on a Hilbert space. An operator T can be decomposed into \(T=UP\) where U is a partial isometry and \(P=| T| =(T^*T)^{1/2}\) with \(N(U)=N(P)\), where N(X) denotes the kernel of an operator X. The following theorem is closely related to the Fuglede-Putnam theorem.
Theorem 1. Let \(T_ k=U_ kP_ k\) be the polar decompositions of \(T_ k\) for \(k=1,2\) and 3. Then the following conditions are equivalent.
(A) \(T_ 1T_ 2=T_ 2T_ 3\) and \(T^*_ 1T_ 2=T_ 2T^*_ 3.\)
(B) (1) \(P_ 3P_ 2=P_ 2P_ 3\), (2) \(P_ 1U_ 2=U_ 2P_ 3\), (3) \(U_ 3P_ 2=P_ 2U_ 3\), (4) \(U_ 1U_ 2=U_ 2U_ 3\) and (5) \(U^*_ 1U_ 2=U_ 2U^*_ 3.\)
Theorem 2. Let T be normal. Then there exists a unitary operator U such that \(T=UP=PU\) and both U and P commute with \(V^*\), V and \(| A|\) of the polar decomposition \(A=V| A|\) of any operator commuting with T and \(T^*.\)
Theorem 2 yields the following well-known result by F. Riesz an B. Sz.-Nagy [Leçons de l’analyse fonctionnelle (1965; Zbl 0122.112)].
Theorem A. Every normal operator T can be written in the form UP where P is positive and U may be taken to be unitary and such that U and P commute with each other and with all operators commuting with T and \(T^*.\)
Corollary 1. Let \(T_ k=U_ kP_ k\) be the polar decompositions of \(T_ k\) for \(k=1,2\) and 3 and let \(T_ 1T_ 2=T_ 2T_ 3\) and \(T^*_ 1T_ 2=T_ 2T^*_ 3\). Then
(1) \(\overline{R(T_ 2)}\) reduces \(U_ 1,P_ 1\) and \(T_ 1\); \(N(T_ 2)\) reduces \(U_ 3,P_ 3\) and \(T_ 3.\)
(2) \(U_ 1| \overline{R(T_ 2)}\) (resp. \(P_ 1| \overline{R(T_ 2)}\), \(T_ 1| R\overline{(T_ 2))}\) is unitarily equivalent to \(U_ 3| N(T_ 2)^{\perp}\) (resp. \(P_ 3| N(T_ 2)^{\perp},T_ 3| N(T_ 2)^{\perp}).\)
(3) When \(T_ 2\) has dense range, then if \(U_ 3\) (resp. \(P_ 3\) and \(T_ 3)\) has an algebraic definite property \(\Sigma\) with polynomials \(\{P_{\alpha}\}\), then so has \(U_ 1\) (resp. \(P_ 1\) and \(T_ 1).\)
(4) When \(T_ 2\) is injective, then if \(U_ 1\) (resp. \(P_ 1\) and \(T_ 1)\) has an algebraic definite property \(\Sigma\) with polynomials \(\{P_{\alpha}\}\), then so has \(U_ 3\) (resp. \(P_ 3\) and \(T_ 3)\).
Theorem 1. Let \(T_ k=U_ kP_ k\) be the polar decompositions of \(T_ k\) for \(k=1,2\) and 3. Then the following conditions are equivalent.
(A) \(T_ 1T_ 2=T_ 2T_ 3\) and \(T^*_ 1T_ 2=T_ 2T^*_ 3.\)
(B) (1) \(P_ 3P_ 2=P_ 2P_ 3\), (2) \(P_ 1U_ 2=U_ 2P_ 3\), (3) \(U_ 3P_ 2=P_ 2U_ 3\), (4) \(U_ 1U_ 2=U_ 2U_ 3\) and (5) \(U^*_ 1U_ 2=U_ 2U^*_ 3.\)
Theorem 2. Let T be normal. Then there exists a unitary operator U such that \(T=UP=PU\) and both U and P commute with \(V^*\), V and \(| A|\) of the polar decomposition \(A=V| A|\) of any operator commuting with T and \(T^*.\)
Theorem 2 yields the following well-known result by F. Riesz an B. Sz.-Nagy [Leçons de l’analyse fonctionnelle (1965; Zbl 0122.112)].
Theorem A. Every normal operator T can be written in the form UP where P is positive and U may be taken to be unitary and such that U and P commute with each other and with all operators commuting with T and \(T^*.\)
Corollary 1. Let \(T_ k=U_ kP_ k\) be the polar decompositions of \(T_ k\) for \(k=1,2\) and 3 and let \(T_ 1T_ 2=T_ 2T_ 3\) and \(T^*_ 1T_ 2=T_ 2T^*_ 3\). Then
(1) \(\overline{R(T_ 2)}\) reduces \(U_ 1,P_ 1\) and \(T_ 1\); \(N(T_ 2)\) reduces \(U_ 3,P_ 3\) and \(T_ 3.\)
(2) \(U_ 1| \overline{R(T_ 2)}\) (resp. \(P_ 1| \overline{R(T_ 2)}\), \(T_ 1| R\overline{(T_ 2))}\) is unitarily equivalent to \(U_ 3| N(T_ 2)^{\perp}\) (resp. \(P_ 3| N(T_ 2)^{\perp},T_ 3| N(T_ 2)^{\perp}).\)
(3) When \(T_ 2\) has dense range, then if \(U_ 3\) (resp. \(P_ 3\) and \(T_ 3)\) has an algebraic definite property \(\Sigma\) with polynomials \(\{P_{\alpha}\}\), then so has \(U_ 1\) (resp. \(P_ 1\) and \(T_ 1).\)
(4) When \(T_ 2\) is injective, then if \(U_ 1\) (resp. \(P_ 1\) and \(T_ 1)\) has an algebraic definite property \(\Sigma\) with polynomials \(\{P_{\alpha}\}\), then so has \(U_ 3\) (resp. \(P_ 3\) and \(T_ 3)\).
MSC:
| 47B20 | Subnormal operators, hyponormal operators, etc. |
| 47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |
