Sebestyén, Zoltán

English

Sebestyén Zoltán

University of Szeged

Acta Sci. Math. 4 6 ( 1 9 8 3 ) , 2 9 5 — 2 6 0 On ranges of adjoint operators in Hilbert space ZOLTÁN SEBESTYÉN Introduction Let A be a given densely defined operator in the (complex) Hilbert space H. Let further y and z be elements in H. The relation (1) (Ax,z) = (x,y) [xd2)(A)), where Si (A) stands as usual for the domain of A, is fundamental for the definition of A*, the adjoint of A. Namely, z is in 3>(A*) if sup {|(Ax, z)|: x€9(A), ||x|| S l } < = o holds, that is by the Riesz Representation Theorem if and only if there is an y in H satisfying (1). The reverse problem is the characterization of Sfc(A*), the range of A* : y is in M(A*) if there is an element, z in 3(A*) for which (1) holds. We shall show that this is the case if and only if sup {|(*. y)\: X€9(A), |M*|| 1} < -holds (Theorem 1). As an application we obtain results concerning the factorization of a given densely defined operator C in H in the form CcA*B by which we mean that B is an operator in H defined at least on 3>(C), and for any x in @>(C), Bx£ €@(A*) and Cx=A*(Bx). In general, as a Zorn's argument shows, &(A*)=>&(C) is sufficient for such a factorization, but we produce a minimal B in the sense that (2) 115x11 || k|| for x£3(C), u£$(A*); Cx = A*u. The question of the boundedness of B is also analyzed in the hope that we shall be able to answer the question raised by R. G. DOUGLAS [1] concerning the factoriza-tion of unbounded operators, especially with a bounded cofactor. Our constant reference is [2]. Received March 5, 1982, and in revised form December 14, 1982. 2 9 6 Z o l t á n S e b e s t y é n Results Theorem 1. Let y and A be a unit vector and a densely defined operator, respectively, in a Hilbert space H. The following two assertions are equivalent: (i) There exists a unique vector z in H such that (3) y = A*z and Hz|| s Hu|| for u£3)(A*), y = A*u. (ii) My := sup {\(x, y)\: x£®(A), \\Ax\\ S l } < » . lf(i) and (ii) are valid, then My=\\z\\. Proof, (ii) simply follows from (i) since for any x in S>(A), \(x, y)| = \(x, A*z)| = |(Ax, z)| ^ || z|| • \\Ax\\; we see also that ilij,^||z||. (ii) implies (i): Assuming (ii) we have \(x, y)\^MylAx\ for any x in 2!(A). So the map Ax>-»(x, y) is a bounded linear functional on 31(A). It has a unique bounded linear extension to 01(A), the norm closure of 01(A). By the Riesz Repre-sentation Theorem there exists a unique vector, z, in ,<%(A) for which (1) holds. Then z is in 2>(A*) and y=A*z. If ujiz is from 3>(A*) and y—A*u, then (Ax, z)=(x, A*z)=(x, A*u) = —(Ax,u) for every x in 3(A). Since z is, while u is not in &t(A), it follows that Ml = sup {\(Ax, z)|: X€9(A), |M*|| ^ 1} = = sup {|04*, «)|: X€0(A), №'*H s l } < ll«|| Thus (3) holds and the z with this property is unique. The proof is complete. Theorem 2. Let A and C be densely defined operators in a Hilbert space H. The following three assertions are equivalent: (i) There exists an operator B in H such that (4) CczA*B and B fulfils (2). (ii) @(C)a@(A*). (iii) My(C):=sup {l(x,Cy)l: x£®(A), M*||=SlH~ (y€9(£)). Proof, (i) clearly implies (ii). Further (ii) implies (iii) since for any y in 3>(C) there exists (by assumption) a u in S}(A*) such that Cy=A*u whence for any JC in @(A), \(x, Cy)| = \(x, A*u)\ = |(Ax, u)| S ||u|| • \\Ax||, and thus (iii) follows. Lastly assume (iii) and prove (i). For a fixed x in 3i(C) there exists, by Theorem 1, a unique vector z in H such that (3) holds with y-Cx. Writing O n r a n g e s o f a d j o i n t o p e r a t o r s i n H i l b e r t s p a c e 2 9 7 z=Bx we get just (2) as desired. We have to show only that Bx is a linear function of x if x varies on $>(C). Recall that, as the proof of Theorem 1 indicates, Bx is in M(A) for any x in ©(C). Thus if x, x' are arbitrary vectors from 2(C), for any y belonging to S>(A) we have 0 - (C(J<+x'), y)—(Cx, y)-(Cx' y) = (A*B(x+x'), y)-(A*Bx, y)-(A*Bx', y) = = (B(x+x'), Ay)~(Bx, Ay)—(Bx', Ay) = (B(x+x')-Bx+Bx', Ay), which shows that B(x+x')—Bx+Bx'. The proof of B(lx)=XBx for a scalar X is similar. The proof is complete. The following is analogous to [1, Theorem 2, (3)] due to Douglas. Corol lary 1. If C of Theorem! is closed then sup {115x11: xe®(C), Ml + 1  Cx|| S 1} ^ oo. In particular, B is bounded if C is. Proof. By assumption, C has a closed graph. Hence we have to show that the linear operator given by {x, Cx)^—Bx (x£@(C)) also has a closed graph.. In other words, assuming that x„—x, Cx„—Cx and Bxn—u, we must conclude u=Bx. Since Cx„—Cx means that A*Bxn->-Cx, by the closedness of A* we get A*u=Cx=A*Bx. But since Bx„ is in 1%(A), u is in 8fc(A), too. As (Ay, u)= = (y,A*u)=(y,A*Bx) = (Ay,Bx) for every y£@(A), it follows that ||M|| = sup {|(Ay, u)\: yd®(A), \\Ay\\ == 1} = = sup {|(Ay, Bx)|: y£0(A), \\Ay\\ § 1 } ^ ||5x||, whence by the uniqueness of Bx we have u=Bx indeed. Remark 1. If in Theorem 2 the operator A is bounded and C is closed, further if we take S>(B)=2>(C), then B is closed. Indeed, if x and Bx„—u, where xne@(C) (n=l, 2, ...), then Cx„=A*Bxn—A*u so that A*u=Cx=A*Bx, and an argument similar to that appearing in the proof of Corollary 1 shows u=Bx. Theorem 3. The following four assertions are equivalent: (i) The operator B in Theorem 2 (i) is bounded. (ii) 52(,4*)=> 52(C) and sup {inf [||z||: z£®(A*), Cy = A*z]: y£®(C), ||j|| == 1} < (iii) sup {\(x, Cy)|: xi®(A), \\Ax|| =§ 1, y€®(C), s l } < » . (iv) ®(C*) z) ® (A) and sup {||C**||: x£9(A), |M*|| S 1} < «,. 2 9 8 Z o l t á n S e b e s t y é n : O n r a n g e s o f a d j o i n t o p e r a t o r s i n H i l b e r t s p a c e Proof. Assume first (i). We know from Theorem 2 that for any y in 3(C), infillrB: ZZ9(A*), Cy = A*z] = \\By\\ =S ||2?|| ||j|| holds. This proves (ii). But (ii) implies (iii) since we know also from Theorem 2 that sup {|(x, Cy)\: x£3(A), \\Ax\\ s l } = inf [||z||: zZ3(A)*, Cy = A*z] for any y in 3(C). For the same reason (iii) implies (i). But (iv) also follows from (iii) since by (iii) 3(A)cz3(C*) and since for any x in 3(A), ||C*x|| = sup y)|: y£3(C), ||j|| s l } = = sup {\(x,Cy)\: y€3(C), | | ^ 1}. Finally (iv) implies (iii) since for any x in 3(A), x is in 3(C*) and |(*> Cy)\ = |(C*x, y)\ S ||C*x|| • ||j|| holds for any y in 3(C). Remark 2. Assuming that A* is densely defined or, what is the same, that A** exists, assertions (i)—(iv) in Theorem 3 are equivalent to (iv)' sup {||C*x||: x€3(A**), |M**x||=Sl}<°°. Indeed, since A** ID A in this case, (iv)' implies (iv). On the other hand, (i) implies now that C*z>(A*B)*^>B*A** and that ||C**|| = \\B*A**x\\ ||.B*|HM***II holds for any x in 3(A**), which proves (iv)'. References [1] R. G. DOUGLAS, On majorization, factorization, and range inclusion of operators on Hilbert spaces, Proc. Amer. Math. Soc., 17 (1966), 413—415. [2] F. RIESZ and B. SZ.-NAGY, Functional Analysis, Ungar (New York; 1960). DEPARTMENT OF MATH. ANALYSIS II EÖTVÖS LORÄND UNIVERSITY MÜZEUM KÖRÜT 6—8 1088 BUDAPEST, HUNGARY 

On ranges of adjoint operators in Hilbert space

http://acta.bibl.u-szeged.hu/14941/1/math_046_295-298.pdf

On ranges of adjoint operators in Hilbert space

Abstract

Similar works

Full text

Available Versions

University of Szeged