SPECTRAL RADIUS ALGEBRAS OF WEIGHTED CONDITIONAL EXPECTATION OPERATORS ON L 2 (F) Groups and Operators, August 2016, Chalmers University, Gothenburg.

Transcription

1 SPECTRAL RADIUS ALGEBRAS OF WEIGHTED CONDITIONAL EXPECTATION OPERATORS ON L 2 (F) Groups and Operators, August 2016, Chalmers University, Gothenburg.

2 The Operators under investigation act on the Hilbert space L 2 (F) and are of the form M w EM u, where M w and M u are (not necessarily bounded) multiplication operators with symbols u and w, and E is the conditional expectation operator relative to σ-subalgebra A. From now on we use the term conditional type operators in stead of weighted conditional expectation operators.

3 The conditional type operators in function spaces were studied by many authors. Here we recall some of them. In the first investigation Moy obtained necessary and sufficient conditions for a linear transformation T between function spaces to be of the form of conditional type operators. Shu-Teh Chen, Moy, Characterizations of conditional expectation as a transformation on function spaces, Pacific J. Math. 4 (1954),

4 Grobler and de Pagter showed that class of partial integral operators are conditional type operators. J. J. Grobler and B. de Pagter, Operators representable as multiplication-conditional expectation operators, J. Operator Theory 48 (2002), P.G. Dodds, C.B. Huijsmans and B. De Pagter, characterizations of conditional expectation-type operators, Pacific J. Math. 141 (1990), J. Herron, Weighted conditional expectation operators, Oper. Matrices 1 (2011),

5 Also, we investigated some classical properties of conditional type operators on L p -spaces. Y. Estaremi and M.R. Jabbarzadeh, Weighted lambert type operators on L p -spaces, Oper. Matrices 1 (2013), Y. Estaremi, Some classes of weighted conditional type operators and their spectra, Positivity. 19 (2015) Y. Estaremi, Unbounded weighted conditional expectation operators, Complex Anal. Oper. Theory 10 (2016)

6 For a σ- subalgebra A of F, the conditional expectation operator associated with A is the mapping f E A f, defined for all non-negative measurable function f as well as for all f L 2 (F), where E A f, by the Radon-Nikodym theorem, is the unique A-measurable function satisfying fdµ = E A fdµ, A A. A As an operator on L 2 (F), E A is idempotent and E A (L 2 (F)) = L 2 (A). We will often write E for E A. A

7 This operator will play a major role in our work and we list here some of its useful properties: If g is A-measurable, then E(fg) = E(f )g. If f 0, then E(f ) 0; if E( f ) = 0, then f = 0. E(fg) (E( f 2 )) 1 2 (E( g 2 )) 1 2. For each f 0, z(e(f )) is the smallest A-set containing z(f ), where z(f ) = {x X : f (x) 0}. A detailed discussion and verification of most of these properties may be found in M. M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993.

8 Here we will briefly review some basic facts about spectral radius algebras. Let H be a Hilbert space, T B(H) and r(t ) be the spectral radius of T. For m 1 we define R m (T ) = R m := ( n=0 ) 1 2 dm 2n T n T n, (1) where d m = 1 1/m+r(T ). Since d m 1/r(T ), the sum in (1) is norm convergent and the operators R m are well defined, positive and invertible.

9 Let B T be the set of all operators S B(H) such that sup R m SRm 1 <. m N Clearly B T is an algebra and contains all operators commute with T.

10 Interested people can find more details in A. Lambert and S. Petrovic, Beyond hyperinvariance for compact operators, J. Functional Analysis, (2005). A. Biswas, A. Lambert and S. Petrovic, On spectral radius algebras and normal operators, In. Univ. Math. J. (2007). A. Biswas, A. Lambert, S. Petrovic and B. Weinstock, On spectral radius algebras, Oper. Matrices. (2008).

11 It appears to be quite difficult to find explicit descriptions of of the operators in B T for a given operator. We now illustrate the level of difficulty one should expect by describing a spectral algebra for a

12 particularly simple operator. Let T = M w EM u be a bounded operator on L 2 (F). Direct computations shows that for every n N(natural numbers) we have T n f = (E(uw)) n 1 we(uf ), T n f = (E(uw)) n 1 ūe( wf ). Hence we get the positive, invertible operator R m for M w EM u as follows: R m = ( ) 1 I + M (E( w 2 ) n=1 d2n m E(uw) 2(n 1) ) M 2 ūem u.

13 It is easy to see that the following equality holds almost every where on X. If we set then we have n=1 d 2n m E(uw) 2(n 1) = v m = d 2 me( w 2 ) 1 d 2 m E(uw) 2, R m = (I + M vmūem u ) 1 2. d 2 m 1 d 2 m E(uw) 2. By an elementary technical method we can compute the inverse of R m as follow: ( Rm 1 = I + M vmū vme( u 2 ) 1 EM u ) 1 2.

14 Here we recall a fundamental lemma in operator theory. Lemma Let T be a bounded operator on the Hilbert space H and λ 0. Then we have λi + T T = λ + T T = λ + T 2. Specially, if T is a positive operator, then λi + T = λ + T.

15 From now on we assume that E( u 2 ) L (A). Now we characterize the spectral radius algebra corresponding to the conditional type operator M w EM u in the next theorem. Theorem Let S B(L 2 (F)). Then S B Mw EM u if and only if N (EM u ) is invariant under S.

16 Therefore we get that there are many different operators that have the same spectral radius algebra. Corollary If M w EM u and M w EM u are bounded operator on the Hilbert space L 2 (F), then B Mw EM u = B Mw EM u. Also we have a sufficient condition for B Mw EM u to be equal to B(L 2 (F)). Corollary If N (EM u ) = {0}, then B Mw EM u = B(L 2 (F)).

17 Proposition If a L 0 (A) such that a 0 and M aū EM u B(L 2 (F)), then M aū EM u B Mw EM u.

18 Every operator T on a Hilbert space H can be decomposed into T = U T with a partial isometry U, where T = (T T ) 1 2. U is determined uniquely by the kernel condition N (U) = N ( T ). Then this decomposition is called the polar decomposition. The Aluthge transformation T of the operator T is defined by T = T 1 2 U T 1 2.

19 Here we recall that the Aluthge transformation of T = M w EM u is T (f ) = χ z 1 E(uw) E( u 2 ) ūe(uf ), f L2 (F). in which z 1 = z(e( u 2 )). Thus T = M w EM u where w = E(uw)ūχz 1 and u = u. We recall that E( u 2 ) r(m w EM u ) = E(uw). Direct computations shows that E(u w ) = E(uw). Hence r(t ) = r( T ). Hence we have the next corollary. Corollary If w and u are positive measurable functions, then T B T where T = M w EM u.

20 Moreover we get that the commutant of M w EM u (in symbol {M w EM u } ) is a proper subset of B Mw EM u when w, u are positive and w u. In the next corollary we get that B T = B T when T = M w EM u and w, u 0. Corollary If T = M w EM u and w, u 0, then B T = B T.

21 Remark Suppose that T = EM u B(L 2 (F)), u L (A) and A, B be σ-subalgebras of F such that A B. If E = E A and S is an operator for which TS = E B ST, then S B T.

22 Here we recall the definition of Q T for T B(H), that is defined in A. Lambert and S. Petrovic, Beyond hyperinvariance for compact operators, J. Functional Analysis, (2005), as follows: Q T = {S B(H) : R m SR 1 m 0}. Q T is a two sided ideal in B T and every operator in Q T is quasi-nilpotent. In the next Theorem we illustrate Q T when T = M w EM u B(L 2 (F)). Theorem Let T = M w EM u and S B(L 2 (F)). Then S Q T if and only if N (EM u ) is invariant under S and N (EM u ) N (S).

23 Now we have an equivalent condition for the spectral radius algebra of a conditional type operator to be equal to B(L 2 (F)). Proposition If T = M w EM u, then B T = B(L 2 (F)) if and only if sup( E( u 2 )v m + E( u 2 )v m m v m E( u 2 ) 1 (1 + E( u 2 )v m )) < where v m = d2 me( w 2 ) 1 d 2 m E(uw) 2.

24 Also, we have an equivalent condition for the conditional type operator M w EM u to be a constant multiple of an isometry. Theorem If T = M w EM u is a bounded operator on the Hilbert space L 2 (F), then T is a constant multiple of an isometry if and only if sup( E( u 2 )v m + E( u 2 )v m m v m E( u 2 ) 1 (1 + E( u 2 )v m )) < where v m = d2 m E( w 2 ) 1 d 2 m E(uw) 2.

25 Recall that for f, g L 2 (F) we can define a rank one operator f g on L 2 (F) by the action (f g)(h) = h, g f for every h L 2 (F), in which, is the inner product of the Hilbert space L 2 (F). In the next proposition we give some conditions under which a rank one operator belongs to the B Mw EM u. Proposition If T = M w EM u and f, g L 2 (F), then f g B T if and only if sup α 1 2 m E(ug) 2 f 2 + v 1 2 m E(uf ) 2 ( g 2 + α 1 2 m E(ug) 2 ) <, m where α m = v m v me( u 2 ) 1.

26 Proposition, A. Lambert and S. Petrovic, 2005 If x, y H are unit vectors, then S B x y if and only if y is an eigenvector for S.

27 Remark For the unit vectors u, v, w of the Hilbert space H we have B u w = B v w. In the next Theorem we describe Q u v for a rank one operator u v in which u, v are in the Hilbert space H. Theorem Let H be a Hilbert space and S B(H). If u, v H, then S Q u v if and only if S = (I P)SP, where P = P H1 and H 1 is the one-dimensional space spanned by v.

28 Let X, Y, Z be Banach spaces. Assume that T B(X, Y ) and S B(X, Z). Then T majorizes S if there exists M > 0 such that Sx M Tx for all x X. Here we recall a Proposition that gives us an equivalent condition for a closed range operator to majorize another bounded operator. Remark Let X be Banach spaces and T, S B(X ) with R(T ) closed. Then T majorizes S if and only if N (T ) N (S).

29 Proposition Let T = M w EM u and u 0. If S Q T and E(u) δ a.e., then EM u majorizes S.

30 Finally, since the rank one operator x y has closed range, the we can obtain the next proposition. Proposition Let x, y H. If T Q x y, then x y majorizes T.

31 Thank You For Your Attention!