E extracta mathematicae Vol. 23, N¶um.3, 235 { 242 (2008) Range, Kernel Orthogonality and Operator Equations Mohamed Amouch Department of Mathematics, Faculty of Science, Semlalia B.O. 2390 Marrakesh, Morocco, [email protected] Presented by Manuel Gonz¶alez Received November 18, 2007 Abstract: Let A be a Banach algebra and L(A) the algebra of all bounded linear operators acting on A. For a; b 2 A, the generalized derivation ±a;b 2 L(A) and the elementary operator ¢a;b 2 L(A) are de¯ned by ±a;b(x) = ax ¡ xb and ¢a;b(x) = axb ¡ x, x 2 A. 2 Let da;b = ±a;b or ¢a;b. In this note we give couples (a; b) 2 A such that the range and the kernel of da;b are orthogonal in the sense of Birkho®. As application of this results we give consequences for certain operator equations inspired by earlier studies of the equation ® + ®¡1 = ¯ + ¯¡1 for automorphism ®; ¯ on Von Neuman algebras. Key words: Elementary operators, orthogonality, operator equation. AMS Subject Class. (2000): 47A20, 47B30, 47B47, 47B15. \ to my wife Hasna" 1. INTRODUCTION Let A be a Banach algebra and L(A) the algebra of all bounded linear operators acting on A. For a; b 2 A, the generalized derivation ±a;b 2 L(A) and the elementary operator ¢a;b 2 L(A) are de¯ned by ±a;b(x) = ax ¡ xb and ¢a;b(x) = axb ¡ x, x 2 A. If La is the left multiplication by a and Rb is the right multiplication by b de¯ned by La(x) = ax and Rb(x) = xb, x 2 A, then ±a;b = La ¡ Rb and ¢a;b = LaRb ¡ I. Let da;b = ±a;b or ¢a;b. Then the following implications hold for a general bounded linear operator T on a normed linear space V , in particular for T = da;b : Ker T ? Im T ) Ker T \ Im T = f0g ) Ker T \ Im T = f0g: Here Im T denote the closure of the range of T and Ker T ? Im T denotes that the kernel of T is orthogonal to the range of T in the sense of Birkho®. 235 236 m. amouch Recall that if M; N are linear subspaces of a normed linear space V , then M ? N in the sense of Birkho® [5] if ka + bk ¸ kak for all a 2 M and b 2 N: Orthogonality of matrices (more generally bounded linear operators on an in¯nite dimensional Hilbert space), in particular orthogonality of the range and the kernel of elementary operators has been studied in recent years. In the ¯rst part of this work we study orthogonality of the range and the kernel 2 of da;b, and we give couples (a; b) 2 A such that Im da;b \ Ker da;b = f0g : Consider the equation ® + ®¡1 = ¯ + ¯¡1 (*) for automorphisms ®; ¯ on Von Neumann algebras. As application of results given in the ¯rst section for elementary operators we study the equation (*) in the setting of Banach algebra. 2. Elementary operators We begin this section by the following lemma. Lemma 2.1. Let a; b 2 A with cb = e. Then Rb(Im ±a;c \ Ker ±a;c) = Im ¢a;b \ Ker ¢a;b : In particular if Im ±a;c \ Ker ±a;c = f0g then Im ¢a;b \ Ker ¢a;b = f0g: Proof. First, observe that if cb = e, then Rb±a;c = ¢a;b. Indeed, for all x 2 A we have that Rb±a;c(x) = axb ¡ xcb = axb ¡ x = ¢a;b(x). Assume that t 2 Rb(Im ±a;c \ Ker ±a;c). Since Rb±a;c = ¢a;c and Rb is continuous for the uniform norm, then t 2 Im ¢a;b \ Ker ¢a;b. Conversely, since Rc is continuous for the uniform norm, then by the same argument we prove that if t 2 Im ¢a;b \ Ker ¢a;b, then t 2 Rb(Im ±a;c \ Ker ±a;c). Remark 2.2. Let a; b 2 A with cb = e. By an easy adaptation of the proof of Lemma 2.1, we get that Rb(Im ±a;c \ Ker ±a;c) = Im ¢a;b \ Ker ¢a;b : range, kernel orthogonality and operator equations 237 2.1. Normal elements. Let a 2 A. The algebraic numerical range V (a) of a is de¯ned by [2]: n o 0 V (a) = f(a): f 2 A and kfk = jf(e)j = 1 where A0 is the dual space of A and e is the identity on A. If V (a) ⊆ R, then a is called hermitian. If n = h + ik where h; k are hermitian elements and hk = kh; then n is called normal. De¯ne © 2 ª D(A) = (a; b) 2 A : Im ±a;b ? Ker ±a;b : In this section we show that normal elements are in D(A). Lemma 2.3. If h; k 2 A are hermitian elements, then so is ±h;k. Proof. Because V (±h;k) ⊆ V (Lh) ¡ V (Lk) = V (h) ¡ V (k) ⊆ R. Remark 2.4. If X is a Banach space, it is known [7] that V (±h;k) = V (h) ¡ V (k) for all h 2 L(X) : So the converse of Lemma 2.3, is true if A = L(X) where X is a Banach space. Lemma 2.5. If m; n 2 A are normal elements, then so is ±m;n. Proof. Assume that m = h + ik and n = p + iq where h; k; p and q are hermitian elements in A such that hk = kh and pq = qp. Then ±m;n = ±h;p + i±k;q with ±h;p±k;q = ±k;q±h;p. Since h; p and k; q are hermitian, then by Lemma 2.3, ±n;p and ±k;q are hermitian. So ±m;n is normal. Theorem 2.6. ([3]) Let E be a Banach space and T 2 L(E). If T is a normal operator then Ker T ? Im T: Theorem 2.7. If m; n 2 A are normal elements, then Ker ±m;n ? Im ±m;n : Proof. Assume that m; n 2 A are normal elements. Then by Lemma 2.5, ±m;n is normal and by Theorem 2.6, Ker ±m;n ? Im ±m;n. 238 m. amouch Corollary 2.8. If a; b 2 A are normal and there exist c 2 A such that bc = e, then Ker ¢a;c \ Im ¢a;c = f0g : Proof. If a; b are normal elements, then by Theorem 2.7 Ker ±a;b ? Im ±a;b. This implies that Ker ±a;b \Im ±a;b = f0g. Using Lemma 2.1, we conclude that Ker ¢a;c \ Im ¢a;c = f0g. In the following theorem we give other couples (a; b) 2 A2 which are in D(A). The technique used to prove this theorem was published ¯rst in [4]. Theorem 2.9. Let a and b in A with bc = e and kck · 1 for some c in A. If kank · 1 and kbnk · 1 for all n 2 N, then Ker ±a;b ? Im ±a;b : Proof. Since we have that nX¡1 anx ¡ xbn = an¡i¡1(ax ¡ xb)bi ; i=0 then nX¡1 anx ¡ xbn ¡ an¡i¡1(ax ¡ xb ¡ y)bi = nybn¡1 ; i=0 n¡1 where y 2 Ker ±a;b: If we multiply this equality right by c we obtain nX¡1 ny = anxcn¡1 ¡ xb ¡ an¡i¡1(ax ¡ xb ¡ y)bicn¡1 ; i=0 so ° ° 1 n° °° °° ° ° °° °o °y° · °an°°x°°c°n¡1 + °x°°b° n 1 nX¡1 ° °° °° °° ° + °an¡i¡1°°ax ¡ xb ¡ y°°bi°°c°n¡1 ; n i=0 hence ° ° 2 ° ° 1 nX¡1 ° ° °y° · °x° + °ax ¡ xb ¡ y° : n n i=0 range, kernel orthogonality and operator equations 239 Passing to the limit n ! 1 we obtain that kyk · kax ¡ xb ¡ yk : Finally, Im ±a;b ? Ker ±a;b. Theorem 2.10. Let a and b in A with kak · 1 and kbk · 1. Then Im ¢a;b ? Ker ¢a;b : Proof. If kak · 1 and kbk · 1, then kMa;bk · 1. So V (Ma;b) ⊆ D where D is the closed unit disk. Since ¢a;b = Ma;b ¡ 1, then V (¢a;b) ⊆ f¸ 2 C : j¸ + 1j · 1g : If 0 is an eigenvalue of ¢a;b, then 0 2 V (¢a;b). Hence 0 is in the boundary of V (¢a;b). It follows from a result of Sinclair [8, Proposition 1] that Im ¢a;b ? Ker ¢a;b. If 0 is not an eigenvalue of ¢a;b then Ker ¢a;b = f0g and the result follows because all linear subspaces are orthogonal to f0g. 2.2. Normal-equivalent elements. Recall that an element a 2 A is hermitian-equivalent if supfk exp(iat)k : t 2 Rg < 1 ; and that n is normal-equivalent if n = r + is where rs = sr and r; s are hermitian-equivalent: we write n¤ = r ¡ is for such an n. De¯ne © 2 ª B(A) = (a; b) 2 A : Im ±a;b \ Ker ±a;b = f0g : Note that every element of D(A) is in B(A). Hence normal elements are in B(A). In this section we show that normal-equivalent elements are in B(A). Definition 2.11. ([1]) Let A be an algebra with involution and a; b 2 A. (a; b) is said to be a couple of Fuglede if ax = xb implies a¤x = xb¤ for every x 2 A. Theorem 2.12.