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General explicit expressions for intertwining operators and direct rotations of two orthogonal projections

In this paper, based on the block operator technique and operator spectral theory, the general explicit expressions for intertwining operators and direct rotations of two orthogonal projections have been established. As a consequence, it is an improvement of Kato's result (Perturbation Theory of Linear operators, Springer-Verlag, Berlin/Heidelberg, 1996); J. Avron, R. Seiler and B. Simon's Theorem 2.3 (The index of a pair of projections, J. Funct. Anal. 120(1994) 220-237) and C. Davis, W.M. Kahan, (The rotation of eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7(1970) 1-46).


Introduction
Let H be a Hilbert space and B(H) the space of all bounded linear operators on H. An operator P is called an orthogonal projection if P = P * = P 2 . Let P be the set of all orthogonal projections in B(H). As well-known, orthogonal projections on a Hilbert space are basic objects of study in operator theory (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and therein references). Orthogonal projections appear in various problems and in many different areas, pure or applied. In this paper, we will pay attention on the characterization to intertwining operators and direct rotations of two orthogonal projections. Let the set of all unitaries in B(H) be denoted by U (H). If P and Q are orthogonal projections and there exists a unitary U ∈ U (H) such that U P = QU, then U is called an outer intertwining operator of P and Q. The set of all outer intertwining operators of a pair (P, Q) of orthogonal projections is denoted by Similarly, if then U is called an inner intertwining operator of P and Q. The set of all inner intertwining operators of a pair (P, Q) of orthogonal projections is denoted by innU Q (P ).
Moreover, if both of P U = U Q and U P = QU hold, then U is called an intertwining operator of P and Q. The set of all intertwining operators of a pair (P, Q) of orthogonal projections is denoted by intU Q (P ).
For a pair (P, Q) of orthogonal projections. A unitary U ∈ U (H) is called a direct rotation from P to Q (see [1] and [10]) if where K ⊥ = I − K if K is an orthogonal projection. If P and Q are orthogonal projections with P − Q < 1, Kato in [13] verified that there exists U ∈ U (H) such that P U = U Q. Moreover, Avron, Seiler and Simon ([6]) proved that if P and Q are orthogonal projections on H with P − Q < 1, then there exists a unitary U ∈ U (H) with U P U * = Q, U QU * = P. If P and Q are orthogonal projections have no common eigenvectors, the mine result shown by Amrein, Sinha ( [2]) implies that there exists a self-adjoint intertwining operator of (P, Q). For a pair (P, Q) of orthogonal projections, we ( [19]) provided a sufficient and necessary condition that there exists an intertwining operator of (P, Q). More recently, Simon ([18]) presented a more elegant proof of our previous result. In the present paper, we will give another alternative proof of the sufficient and necessary condition for the existence of intertwining operator of (P, Q). The proof is more geometrical compared with the proof in [18], and we believe the block operator technique used here has meaning in itself.
For the sake of convenience, we need some notation and terminologies. For A ∈ B(H), the range, the null space, the spectrum, the real part and the adjoint of A denote by R(A), N (A), σ(A), ReA and A * , respectively. A is said to be positive if (Ax, The following lemma is a starting point and a very useful tool in the sequel. Lemma 1.1. ( [11], [19]) If W and L are two closed subspaces of H and P and Q denote the orthogonal projections on W and L, respectively, then P and Q have the operator matrices and , Q 0 is a positive contraction on H 5 , 0 and 1 are not eigenvalues of Q 0 , and D is a unitary from H 6 onto H 5 . I i is the identity on H i , i = 1, . . . , 6.
Remark 1.2. From Lemma 1.1, we will get more information involving with geometry structure between P and Q. For example, (1) Since DD * = I 5 and D * D = I 6 , it implies that dim H 5 = dim H 6 , where dim M denotes the dimension of a subspace M.
(2) If 0 ( or 1) ∈ σ(Q 0 ), then 0 ( or 1) is a limit point of σ(Q 0 ) and 0 / ∈ σ p (Q 0 ), where σ p (T ) denotes the point spectrum of T. In this case, dim 12]) called that the pair (P, Q) is in the generic position. If two orthogonal projections are in the generic position, then H = H 5 ⊕ H 6 and the operator matrices (5) and (6) of P and Q can be simplified as follows respectively. In general, for a pair (P, Q) of orthogonal projections with operator matrices as (5) and (6), denote P and Q by the pair ( P , Q) as the restriction of (P, Q) on H 5 ⊕ H 6 is called the generic part of (P, Q). Let us give a brief outline of the contents of this paper. The general explicit expressions for outer intertwining operators and intertwining operators of two orthogonal projections in the generic position are stated in Section 2. In Section 3, based on block operator technique and spectral theory we give an alternative proof of the sufficient and necessary condition that there exists an intertwining operator of a pair (P, Q). In view of the proof, we get general explicit expressions of intertwining operators of a pair (P, Q). In Section 4, we provide an alternative proof of the sufficient and necessary condition which there exists a direct rotation of a pair (P, Q) and obtain the general explicit expressions of all direct rotations for a pair (P, Q).

General explicit expressions of intertwining operators for a pair (P, Q) in the generic position
For outer (or inner ) intertwining operators and intertwining operators of a pair of orthogonal projections, we have: Theorem 2.1. Let P and Q be two orthogonal projections in the generic position and P and Q have operator matrix forms (7). Then Proof. We define an operator W 0 by the operator matrix with the decomposition H = H 5 ⊕H 6 . By direct computation, W 0 is a unitary on H with W P = QW , and hence W 0 ∈ intU Q (P ) ⊂ outU Q (P ).
Note that for any V ∈ outU Q (P ), W * 0 V is a unitary commutes with P , and for any V ∈ intU Q (P ), W * 0 V is a unitary commutes with both P and Q. We obtain Let U ∈ U (H) and U has the operator matrix with the decomposition H = H 5 ⊕ H 6 . If U P = P U , then U 12 = U 21 = 0, and U has the operator matrix in which U 11 ∈ U (H 5 ), U 22 ∈ U (H 6 ). Moreover, if U P = P U and U Q = QU , then we get Observing that Q 0 ,I 5 −Q 0 on H 5 are injective from Remark 1.2 and U 11 commutes with Q in which U 11 ∈ U (H 5 ), U 11 Q 0 = Q 0 U 11 . By (9), (10) and (12), we see that W 0 U in outU Q (P ) and intU Q (P ) has the operator form given in (a), (b), respectively. The proof is completed. Corollary 2.2. Let a pair (P, Q) of orthogonal projections be in the generic position, and U ∈ intU Q (P ) has the operator matrix form in Theorem 2.

General explicit expression of intertwining operators for two orthogonal projections
In this section, we will devote to general explicit expressions for intertwining operators of two orthogonal projections if there exists an intertwining operator for the two orthogonal projections.
Let P and Q be two orthogonal projections and have operator matrices (5) and (6), respectively. For the pair (P, Q) of orthogonal projections, if the generic part of (P, Q) is ( P , Q) as operator matrices (8), then the pair ( P , Q) as a pair of orthogonal projections on H 5 ⊕ H 6 is in the generic position.
The main goal in this section is to prove the following theorem.
Proof. " ⇒ ". If there exists a unitary U ∈ U (H) such that P U = U Q and U P = QU, then Denote A = P − Q. Then A is a self-adjoint contraction. If U has the operator matrix  (14), we get U A = −AU. Moreover, by (15) and (16), we obtain Comparing two sides of (17) and observing that A 0 , A 0 − I 0 and A 0 + I 0 are injective and dense, it is derived that U 12 = 0, U 13 = 0, U 14 = 0, U 21 = 0, U 22 = 0, U 24 = 0, U 31 = 0, U 33 = 0, U 34 = 0, U 41 = 0, U 42 = 0, U 43 = 0. Therefore, , we can choose a unitary C 2 from H 3 onto H 2 and a unitary C 3 from H 2 onto H 3 . Define an operator where U 1 is a unitary on H 1 , C 2 is a unitary from H 3 onto H 2 , C 3 is a unitary from H 2 onto H 3 , U 4 is a unitary on H 4 and U 0 is a unitary on H 5 with Q 0 U 0 = U 0 Q 0 , by directly checking, U is a unitary on H and U P = QU and U Q = P U. From the proof above, we have Remark 3.2. Let (P, Q) be a pair of orthogonal projections. From the proof of Theorem 3.1, if dim R(P ) ∩ N (Q) = dim N (P ) ∩ R(Q), then the intertwining operator of the pair of orthogonal projections is not unique. Moreover, it can be chose as a self-adjoint unitary. Even though the intertwining operator can be chose as a self-adjoint operator, it is also not unique by Corollary 2.2.
As a consequence, we give an alternative proof of Theorem 2.2 in [15].
Proof. Let P and Q have operator matrices (5) and (6), respectively. Then Denote the generic part ( P , Q) of (P, Q) as the operator matrices (8). We get By Theorem 2.1, there exists a unitary U on H 5 ⊕ H 6 such that In this case, where Evidently, U is a unitary, and Q = U P U * and P = U QU * . Hence,

General explicit expression of direct rotations on a pair of orthogonal projections
The concept of a direct rotation of a pair on orthogonal projections due to Davis (see [10]).
Moreover, if P and Q have the operator matrices (7), then Proof. If there exists a unitary operator S satisfying (22), then from SP = QS we get Hence, from S 2 = (Q ⊥ − Q)(P ⊥ − P ), we obtain Let P, Q and S have operator matrices with respect to the decomposition H = R(P ) ⊕ N (P ), respectively. From (25), So that, Hence, Moreover, In general, by Theorem 2.1, there exist two unitaries U 0 , V 0 ∈ U (R(P )) such that So that, and Since Q 0 is injective, by (27) and (28), it is clear that U 0 = I R(P ) , V 0 = −I R(P ) . Therefore, by directly checking, S satisfies (22). It is the direct rotation from P to Q. Moreover, if P and Q with dim R(P ) ∩ N (Q) = dim N (P ) ∩ R(Q) have the operator matrices (5) and (6) with respect to the space decomposition H = ⊕ 6 i=1 H i , then where C is an arbitrary unitary from H 3 onto H 2 .
This shows that K 2 is an invariant subspace of S. In this case, S has the operator matrix with respect to the decomposition H = ⊕ 3 i=1 K i . Furthermore, since if y = y 1 + y 2 , where y 1 ∈ R(P ) ∩ N (Q) and y 2 ∈ N (P ) ∩ R(Q), we get S 2 y = S 2 (y 1 + y 2 ) = −y 1 − y 2 = −y. So that S 2 22 y = −y. This means that It implies that S 22 is an invertible operator on K 2 . Furthermore, It follows that S * 23 S 22 = 0. From (32), R(S 22 ) = K 2 , it is derived that S 23 = 0. So that, the operator matrix form (31) can be changed as follows Here, S 22 and S 33 are unitaries on K 2 and K 3 , respectively. If P and Q have the operator matrices (5) and (6), then it is obvious that S 22 as a unitary on K 2 = H 2 ⊕ H 3 has the operator matrix form Conversely, if dim R(P ) ∩ N (Q) = dim N (P ) ∩ R(Q), for any unitary C from H 3 onto H 2 , define an operator S by the form (34), then to directly test the operator S is a unitary which satisfies (22). That is, S is a direct rotation of the pair (P, Q) from P to Q. (2) For a pair (P, Q) of orthogonal projections, if dim R(P ) ∩ N (Q) = dim N (P ) ∩ R(Q) = 0, then the direct rotation from P to Q is not unique. The general expression of direct rotations S from P to Q has the form (29), where C can be chose over all unitaries from H 3 onto H 2 .
(3) For a pair (P, Q) of orthogonal projections with dim R(P ) ∩ N (Q) = dim N (P ) ∩ R(Q), if the set of all direct rotations from P to Q is denoted by S Q (P ), then As the end, we will give an alternative proof of the extremal property in regard to the direct rotation which is due to Davis ( see [1], [10]). The proof used block operator matrices and spectral theory may give us some inspiration in the further study.
Theorem 4.5. Let the pair (P, Q) of orthogonal projections be in the generic position. The direct rotation U from P to Q has the extremal property