Polar decompositions of normal operators in inde¯nite inner product spaces Christian Mehl, Andr¶e C.M. Ran and Leiba Rodman Abstract. Polar decompositions of normal matrices in inde¯nite inner product spaces are studied. The main result of this paper provides su±cient conditions for a normal operator in a Krein space to admit a polar decomposition. As an application of this result, we show that any normal matrix in a ¯nite di- mensional inde¯nite inner product space admits a polar decomposition which answers a±rmatively an open question formulated in [2]. Furthermore, neces- sary and su±cient conditions are given for normal matrices to admit a polar decomposition or to admit a polar decomposition with commuting factors. Mathematics Subject Classi¯cation (2000). 47B50, 46C20, 15A23. Keywords. Krein spaces, polar decompositions, normal operators, inde¯nite inner products. 1. Introduction Let H be a (complex) Hilbert space, and let H be a (bounded) selfadjoint oper- ator on H, which is boundedly invertible. The operator H de¯nes a Krein space structure on H, via the inde¯nite inner product [x; y] = hHx; yi; x; y 2 H; where h¢; ¢i is the Hilbert inner product in H. All operators in the paper are assumed to be linear and bounded. We denote by L(H) the Banach algebra of bounded linear operators on H. The adjoint of an operator X 2 L(H) with re- spect to h¢; ¢i will be denoted by X ¤. A large part of this work was performed while the ¯rst author was visiting the Vrije Universiteit Amsterdam and while he was supported by the Research Training Network HPRN-CT-2000- 00116 of the European Union, \Classical Analysis, Operator Theory, Geometry of Banach Spaces, their interplay and their applications". Research of the third author was supported in part by NSF grant DMS-9988579 and by a Faculty Research Assignment grant from the College of William and Mary. 2 Christian Mehl, Andr¶e C.M. Ran and Leiba Rodman An operator X 2 L(H) is said to be an H-isometry if [Xx; Xy] = [x; y] for all x; y 2 H, and is called H-selfadjoint if [Xx; y] = [x; Xy] for all x; y 2 H. An operator X 2 L(H) is called H-normal if XX[¤] = X[¤]X; where X[¤] is the adjoint of X with respect to the inde¯nite inner product [¢; ¢]. Given a (linear bounded) operator X on H, a decomposition of the form X = UA; where U is an invertible H-isometry (in other words, U is H-unitary) and A is H- selfadjoint, is called an H-polar decomposition of X. An analogous decomposition of the form X = AU will be called a right H-polar decomposition for X. In the context of positive de¯nite inner products, polar decompositions (which are usually taken with the additional requirement that A be positive semide¯nite and the relaxation that U need be a partial isometry only instead of an invertible one) are a basic tool of operator theory. In context of inde¯nite inner products, they have been studied extensively in recent years (see, e.g., [4, 2, 3, 16, 13]), in particular, in connection with matrix computations [7, 8]. Remark 1. An operator X 2 L(H) admits an H-polar decomposition if and only if it admits a right H-polar decomposition. This follows easily from the fact that X = UA = (UAU ¡1)U. Our main result, Theorem 4, is stated and proved in the next section. In par- ticular, it follows from Theorem 4 that for a ¯nite dimensional H every H-normal operator admits an H-polar decomposition, thereby settling in the a±rmative an open question formulated in [2]. In Sections 3 and 4 we apply the main result to other properties that H-normal operators may have in connection with H-polar decompositions, assuming that H is ¯nite dimensional. In particular, we provide necessary and su±cient conditions for normal matrices to admit a polar decom- position or to admit a polar decomposition with commuting factors. 2. The main result In this section, we will provide su±cient conditions for an H-normal operator to admit an H-polar decomposition. The proof of the main result will be based on the following decomposition that is of interest in itself. Lemma 2. Let X 2 L(H), and let QKer X be the orthogonal (in the Hilbert space sense) projection onto Ker X. Assume that the operator QKer X HQKer X jKer X : Ker X ¡! Ker X (1) has closed range. Then there exists an invertible operator P 2 L(H), a Hilbert space orthogonal decomposition H = H0 © H1 © H2 © H0 (2) e Polar decompositions, normal operators, inde¯nite inner products 3 and a Hilbert space isomorphism H14 : H0 ! H0, such that ¡1 Ker (P XP ) = He0 © H1; (3) and with respect to decomposition (2), P ¡1XP , P ¤HP , and P ¡1X[¤]P have the following block operator matrix forms: 0 0 X13 X14 0 0 0 H14 2 0 0 X23 X24 3 2 0 H22 0 0 3 P ¡1XP = ; P ¤HP = ; (4) 0 0 X X 0 0 H 0 6 33 34 7 6 33 7 6 0 0 X X 7 6 H¤ 0 0 0 7 4 43 44 5 4 14 5 and ¡¤ ¤ ¤ ¡¤ ¤ ¡¤ ¤ ¡¤ ¤ H14 X44H14 H14 X24H22 H14 X34H33 H14 X14H14 2 0 0 0 0 3 P ¡1X[¤]P = ; (5) H¡1X¤ H¤ H¡1X¤ H H¡1X¤ H H¡1X¤ H 6 33 43 14 33 23 22 33 33 33 33 13 14 7 6 0 0 0 0 7 4 5 ¡¤ ¤ ¡1 where H14 := (H14) . Moreover, if X is H-normal, then X23 = 0, X43 = 0, and X33 is H33-normal. ? Proof. Let H = G0 © G1 where G0 = Ker X and G1 = (Ker X) . Then with respect to this decomposition, X and H have the forms 0 X12 H11 H12 X = ; H = ¤ : · 0 X22 ¸ · H H22 ¸ b b12 b By the hypothesis, H11 has closedb range, so we mab y furtherb orthogonally decom- pose G = H © H such that with respect to the decomposition H © H © G the 0 0 1 b 0 1 1 operators X and H have the forms 0 0 X13 0 0 H13 X = 2 0 0 X23 3; H = 2 0 H22 H23 3; b 0 0 X H¤ H¤ H 4 b33 5 4 13 23 33 5 where H22 : H1 ! H1 is invertible.b Then setting b I 0 0 ¡1 P1 := 2 0 I ¡H22 H23 3 0 0 I 4 5 implies 0 0 X13 0 0 H13 P ¡1XP = 0 0 X + H¡1H X ; P ¤HP = 0 H 0 : 1 1 2 23 b22 23 33 3 1 1 2 22 3 0 0 X H¤ 0 H 4 b 33 b 5 4 13 33 5 b e 4 Christian Mehl, Andr¶e C.M. Ran and Leiba Rodman Since H is invertible, we obtain that H13 is right invertible. Let H2 = Ker H13, ? H0 = (Ker H13) , and decompose G1 = H2 © H0. Then there exist invertible operators S : H ! H and T : G ! G such that S¤H T = 0 H , where e 0 0 1 1 e 13 14 H14 : H0 ! H0 is a Hilbert space isomorphism. Then setting P2 £= P1¢(S©I¤H1 ©T ), we get e 0 0 X13 X14 0 0 0 H14 ¡1 2 0 0 X23 X24 3 2 0 H22 0 0 3 P XP = e e ; P ¤HP = : 2 2 0 0 X X 2 2 0 0 H H 6 e33 e34 7 6 33 34 7 6 0 0 X X 7 6 H¤ 0 H¤ H 7 4 e43 e44 5 4 14 34 44 5 Finally, setting e e ¤ ¡1 ¤ 1 ¤ ¡1 I 0 ¡(H14) H34 ¡ 2 (H14) H44 2 0 I 0 0 3 P := P P ; 1 2 0 0 I 0 6 7 6 0 0 0 I 7 4 5 we obtain that P ¡1XP and P ¤HP have the form as in (4). A straightforward computation shows that P ¡1X[¤]P has the form (5). Furthermore, 0 0 ¤ ¤ 2 0 0 0 0 3 P ¡1X[¤]XP = : 0 0 ¤ ¤ 6 7 6 0 0 0 0 7 4 5 Now, let X be H-normal, i.e., P ¡1XX[¤]P = P ¡1X[¤]XP . This implies that the ¯rst two operator columns of P ¡1XX[¤]P are zero, i.e., X13 2 X23 3 H¡1X¤ H¤ H¡1X¤ H = 0: (6) X 33 43 14 33 23 22 6 33 7 £ ¤ 6 X 7 4 43 5 Observe that the ¯rst operator matrix in (6) has zero kernel, because of (3). This implies X43 = 0 and X23 = 0. Then comparing the blocks in the (3; 3)-positions of ¡1 [¤] ¡1 [¤] ¡1 ¤ ¡1 ¤ P XX P and P X XP , we obtain X33H33 X33H33 = H33 X33H33X33, i.e., X33 is H33-normal. Next, we state a lemma that is of a general nature. We say that a point ¸ 2 (X), X 2 L(H), is an eigenvalue of ¯nite type if ¸ is an isolated point of the ¡1 ¡1 spectrum (X) and the spectral projection (2¼i) j»j=²(»I¡X) d», where ² > 0 is su±ciently small, has ¯nite rank. It is easy to seeR (by using the decomposition of H as a direct sum of two X-invariant subspaces so that X ¡ ¸I is invertible on one of them, and X ¡ ¸I is nilpotent on the other) that if ¸ is an eigenvalue of Polar decompositions, normal operators, inde¯nite inner products 5 ¯nite type of X, and if M is an X-invariant subspace such that ¸ 2 (XjM), then ¸ is an eigenvalue of ¯nite type of the restriction XjM.