Operator differentiable functions - Springer Link

Integral Equations and Operator Theory Vol. 13 (1990)

0378-620X/90/040461-2751.50+0.20/0 (c) 1990 Birkhguser Verlag, Basel

O P E R A T O R DIFFERENTIABLE FUNCTIONS

J. Arazy, T.J. BartonJ and Y. Friedman i A scalar function f is called operator differentiable if its extension via spectral theory to the self-adjoint members of !B(H) is differentiable. The study of differentiation and perturbation of such operator functions leads to the theory of mappings defined by the double operator integral

# We give a new condition under which this mapping is bounded on ~3(H). We also present a means of extending f to a function on all of ~B(H) and determine corresponding perturbation and differentiation formulas. A connection with the "joint Peirce decomposition" from the theory of JB*-triples is found. As an application we broaden the class of functions known to preserve the domain of the generator of a strongly continuous one-parameter group of *-automorphisms of a C*-algebra.

1

INTRODUCTION

Let A be a C*-algebra and let f : R ~ C be a continuous function. By the usual functional calculus f can be extended to a function, also called f, on the self-adjoint members of A. It is natural to ask when the extended function f is differentiable (we shall always mean GgLteaux differentiable; however, all of the results of which we are aware establish Fr~chet differentiability). This problem has been posed by H. Widom [18] in the book "Linear and Complex Analysis Problem Book." It is not difficult to see that a necessary condition is that the scalar function f must be continuously differentiable. In 1partially supported by NSF grant DMS8603064.

462

Arazy, Barton and Friedman

case A is commutative, it is easily seen that this condition is also sufficient.

However,

in [9] Farforovskaya constructed an example of a C 1 function whose extension to tile self-adjoint operators on a Hilbert space is not differentiable. Daletskii and Krein [7] considered this problem in the context of the self-adjoint operators on a Hilbert space H. They showed that if f is C 2 then its extension is differentiable. Moreover, they obtained a formula for the derivative of f in terms of a notion of "iterated operator integrals", which they also introduced. Birman and Solomyak [2,3,4] refined this concept and introduced and developed the theory of "double operator integrals" which has served as the basis for all subsequent research in this area (see sections 2 and 3). They also found much sharper sufficient conditions for operator differentiability which include, for example, scalar functions whose derivative is Hglder continuous of any order and scalar functions whose derivative has absolutely convergent Fourier series. Still sharper sufficient and necessary conditions were found by Peller [14] (see section 4). In section 2 and in the first part of section 3 we shall present those parts of this theory upon which the present paper depends. We feel it desirable to include such an exposition since the relevant articles either are in Russian or do not enjoy a wide circulation in English. Generally speaking, these authors obtained their results upon careful analysis of the Fourier expansion of the scalar function f. In section 4 we shall instead proceed by considering a decomposition into Mgbius functions. The sufficient condition so obtained is at least as sharp as that of Peller (we don't yet know if it is strictly sharper). This approach is also more natural and elementary than that via Fourier analysis. In section 3 we describe a little known functional calculus which allows the scalar function f to be extended to a function which is defined for all bounded operators on H. We thus expand the scope of the operator differentiability problem to include such extensions, and provide a sufficient condition for differentiability of f in this case. Along the way we obtain a perturbation formula for functions of bounded operators which is more general than any others of which we are aware. The functional calculus just referred to is the usual functional calculus for ~ ( H )


463

considered as a JB*-triple rather than as a C*-algebra (see [15] and the references cited there for further information on JB*-triples). A fundamental tool in the theory of JB*triples is the joint Peirce decomposition. We show that, even in the self-adjoint case, the derivative of f is computed by a Schur multiplication relative to this decomposition. In section 5 we exploit a connection between the theory of operator differentiation and the study of dynamical systems. In the C*-algebraic formulation of quantum mechanics, the time evolution of a quantum mechanical system is modelled by a strongly continuous one-parameter group of *-automorphisms. We simultaneously consider such systems as well as those whose time evolution is given by a strongly continuous one-parameter group of isometries. The generator of such a flow is always a (possibly unbounded) derivation of the JB*-triple structure on the C*-algebra. (Derivations have received extensive study in the context of C*-algebras, see [5]). One aspect of the study of generators of these systems centers on the problem of determining those scalar functions f whose extensions leave the domain of a generator invariant. We shall apply the foregoing material to determine a broad class of functions with this property. This result generalizes a theorem of Powers (as amended by Bratteli and Robinson, see [6, Theorem 3.2.32]). Although we shall work in the context of ~ ( H ) , where the theory of double operator integrals is available to us, all of the statements regarding differentiability remain true for general C*-algebras courtesy of the Gelfand-Naimark Theorem (applied to the bidual). However, the formulas for calculating perturbations and derivatives are meaningful only for ~ ( H ) since they are expressed in terms of double operator integrals.

2

DOUBLE O P E R A T O R INTEGRALS AND SCHUR MULTIPLIERS

In this section we shall review the construction of the double operator integral and survey some of the related theory needed in the sequel. Throughout this article, let H be a fixed complex Hilbert space. Let A be a set. A countably additive function E(.) defined on a a-algebra of subsets of A taking values in the set of orthogonal projections on some Hilbert space is

464


an orthogonal measure on A. Thus, an orthogonal measure is a spectral measure (see [8]) except that the condition E(A) = 1 need not be met. Let E and F be orthogonal measures on A and M taking values in the set of orthogonM projections on H.

We construct a product measure from E and F in the

following way. For each measurable rectangle A1 = A1 • M1 C A • M and for each elementary tensor ~ | q in the Hilbert space H | H define G ( A , ) ( ( | 77) = E ( A , ) ( @ F(M1)~,

(2.1)

By standard measure theoretic techniques G can be extended to an orthogonal measure on H @ H . Identifying H |

with the Hilbert-Schmidt class C2 = C2(H), it follows from

(2.1) that G(A1)x = F(M1)xE(A1) for all z E C:. From the spectral theorem for normal operators it follows that

r~ f f r

#)dG

is an isometric mapping from the G-essentiMly bounded functions % : A •

(2.2) M --+C into

the normal operators on C2. D e f i n i t i o n 2.1 For x 6 C2 define the double operator integral

9(x): / / r

(2.3)

h M

to be the value of the operator on C2 defined by (2.2) applied to x. We would like to determine when r can be extended to all of ~8(H). r

is

defined for all rank one operators x on H, and so one can determine if sup{lie(x)[[1 ] Ilx[[1 < 1} is finite, where H" [11 is the trace class norm. If so, then r is bounded on the trace class C1. Since r replaced by r

I]r

considered as on operator on C2, is given by (2.3) with r

is also finite and so r can be extended to a bounded operator on

~ ( H ) by duality. D e f i n i t i o n 2.2 A function r : A • M ~ C is a Schur multiplier (with respect

to E and F ) i[ (2.3) defines a bounded operator on fS(H). The class of Schur multipliers is denoted by ~Yt(E,F). / ] 6 6 fl3~(E,F) then its multiplier norm IIeH'~,(E,r) i~

llr

hrazy, Barton and Friedman

465

The intersection of the classes 9J~(E, F ) over all orthogonal measures E on' A a n / F on M is denoted ffJ~(A,M). We shall also write ffJl(A) = 9~(A, A). In [4], Birman and Solomyak introduced the class of admissible functions, defined as follows. Let (gt, dw) be a measure space and denote by L~(A, L2(Ft)) the space of all functions a : A • ~t --~ C satisfying

/

11

L~(A)

Similarly define L ~ ( M , L2(F/)) to be the space of all fl : M x ~ --~ C such that

c(~) =

~1/2

I~("~)l~d~ L~(M)
0.

g r 9 ~s(R) and if both IIx. - xoll --* 0 a ~ d ItYm - Y011 --* 0, t h e ~ Ilem,n - O0,01l --* 0, In fact, a far more general result is proved in [4]. Notice that Theorem 3.1 allows the double operator integral to be considered in the Riemann-Stieltjes sense, for certain r Also notice that the functions with which we shall be concerned, namely m~ and the function m~- to be defined below, are symmetric in their variables.

468


Theorem 3.1 can be applied to the problem in (3.2) to obtain the following differentiation formula: if m ] E gis(R), then f i3 differen~iable and

f'(x).y = f f m;(A, I~)E(dp)yE(dA). Ii

(3.3)

R

Similar results also hold for unitary operators on H, but we shall not give any details here as they can be obtained from the material in section 5.

3.2 Extension to bounded operators with the triple functional calculus. ~ ( H ) , like all C*-algebras (and some other spaces), has another algebraic structure induced by the Jordan triple product

{xyz} - zy*z + zy*z 2 With this ternary product fl3(H) becomes a JB*-triple (see [151 and the references therein for further background on JB*-triples). The principal advantage in viewing ~3(H) in this way is that each element x has a "tripotent"-valued spectral resolution (defined below) analogous to the spectral resolution of self-adjoiat operators used in the preceding paragraphs. The differentiation formula we obtain below is also expressed in terms of a fundamental structure in JB*-triples, the joint Peirce decomposition (rather, a continuous version thereof). We shall, however, only use standard techniques so that no knowledge of JB*-triples is required. We shall use the following notations: [t+ = (0, oo), R~ = [0, oc), and R_ = (-~,0).

Let x E fl3(H), let x = viz I be its polar decomposition, and let c(.) be tile

spectral measure of Ix[.

For each Borel subset A of R+ define the partial isometry

(tripotent) v(A) = v.e(A). I6 A, and A2 are disjoint norel subsets of R+, then z,t = v(At ) and v2 = v(A2) are orthogonal partial isometries, i.e., v~v2 = 0 = viva. We call v(-) tile

Iripotenr

spectral measure of x. Let f : R+ --* C be any bounded Borel function,

and consider f to be extended to all of R so that it is an odd function. We then define the triple functional calculus by F(x) = f,+ f(~) v(d),) = vf(txl). The idea of basing the spectral resolution on a ternary operation goes back at least as far as the early 1960's to the work of Hestenes ([11,12]).


469

The triple functional calculus is easily seen to extend the self-adjoint functional calculus. Let x be self-adjoint and let E(.) be its spectral measure. Then the polar decomposition x = vlx I is given by v = f. sgn(A)E(dA) and Ixl = f.+ Ae(dA), where e(A) = E(A) + E ( - A ) for all Borel A C R+. Consequently,

F(x)

= / f(A)v(dA) R+

R

R+

R+ =

R_

R_

ff(A)E(dA) R

= f(x). By virtue of the above calculation we shall without ambiguity use the symbol f to denote the extension of the scalar function f in either sense. We shall also realize the triple functional calculus as a restriction of the selfadjoint functional calculus in the following way. Associate to each x E ~ ( H ) the selfadjoint operator

x* 0 e ~(H , H ) .

(3.4)

If v is a partial isometry in ~ ( H ) , then the spectral resolution of ~3is given by ~3= R(v) + 0[ 1 - vv" [

0

where

0 ] 1 - v*v j

vv'v. v.vV]

From this it follows immediately that if v(.) is the tripotent valued spectral measure of x and v = v(R+), then 2 has the spectral measure/~ determined by

~(A) =

R(v(A))

if A C R+

n(v(-A))

if/x c r _

[17v*

1-0]v*v

irA={0}

(3.5)

470


and that

Pf(~)Q = f ( x ) ,

(3.6)

where P and Q are the orthogonal projections P(~,~7) = (~,0) and Q(~,~7) = (0,77). The idea of viewing the triple functional calculus on ~ ( H ) in this way was communicated to one of the authors by U. Haagerup and we thank him for this. We turn now to the perturbation and differentiation formulas for bounded operators. Let x mid y be bounded operators on H with tripotent-valued spectral measures v(.) and w(-). Let v = v(R+) and w = w(ll+), let A C R~., and define spectral measures Lv(A)

:

w(A - { 0 } ) w ( A - {0})* + XA(0)(1 -

n~(A)

:

v ( A - { 0 } ) ' v ( A - {0}) + ~ ( 0 ) ( 1

where Xa denotes the characteristic function of A. sures of

ww')

- v'v)

L y and R ~ are the spectral mea-

(yy*)~/2 and (x'x) ~/2, respectively. Also define the generalized symmetric and

antisymmetric parts of z E ~ ( H ) with respect to v and w by so,~(z)

= ~1( z

+wz*v) and d.,,.(z)

With these notations we formally define the

=

~1( z -

wz*v) .

joint Peirce Schur multiplication operator"

relative to x and y induced by f O~,~(z) =

f f mY(A'P)LY(dp)S"~(z)R~(dA) o o R+X+

+ J/mM~,v)L'(d')A.,-(z)n:(dA) 0 0 R+R+

where m~-(A, #) - f(A) + f(/~) Since f is odd, rn)-(~, #) = mT()~, - # ) , and so the indeterminate form m~-(0, 0) is defined to be if(0). By Theorem 2.4 q),,~ is a bounded operator on ~B(H) if both m)- and m~belong to 9.1(R). Since f is an odd function, the latter requirement is redundant. rationale for the name of this operator is provided by Remark 3.4 below.

A


471

T h e o r e m 3.2 Let x,y E fB(H). If f : R --+ C ia an odd function for which m 7 E Pas(R), then f(~) - f(x) = ~,~(y - x).

(at)

P r o o f : Let E and F be the spectral measures of ~ and ~). Fix n E N, let Ao = {0} = M0, and let Aj = ( s

~lixll I and Mk = ( ~@IiYII, {IlYil 1, 1 _< j, k < n. Choose ),j 9 Aj and

#k 9 Mk and define x~ = ~ )~jvj and y~ = ~ #kwk, where vj = v(Aj) and wk = w(Mk). j=l

k-----1

L e t / ~ a n d / ~ be the spectral measures ofa?~ and ~)~. Then Ha?~-a?ll ---+0 and ll~)~-~)ll ---+ 0 by the spectral theorem, and so by (3.1), Theorem 3.1, and equation (3.6) we have

R

The integral appearing above is in fact a discrete sum, which we compute with the aid

of (3.5).

R

R

= k f(•J) - f(#k)F(Mk)(!) -- &)/~(Aj) j,k=o Aj #k

+ ~ f ( - ) U ) - f(#k),~(Mk)(O _ ~ ) E ( - A j ) j,k=l --'~j -- #k + ~

f(--"~J) -- f(------#k)*~(--Mk)(!) -- &)/~(-Aa)

+ ~ I(-Aj) -~(0)~{Mo)(~ - ~)~(-a,) d=l

k=l

--)U

472


Applying P on the left and Q on the right we obtain

1 ~

f()~r

~-

+ ~ f(xj) - f ( - a j ) ( 1 _ ww')(y j=l

~)v;vj

-2~

+ ~ f(#k) - I(-#k)wkwi(y _ x)(1 - v'v) k=~ 2#k + / ' ( 0 ) ( 1 - ww')(y - x)(1 - v'v) which is equal to ~5 ,y,(y - x).

Since f is odd, from the hypothesis it follows that

rn~ e 9.Is(R) also. Since tlYnY~-YY*I[ --* 0 and Ilx*nxn-x*xtl --~ O, we can apply Theorem 3.1 again to each of the double operator integrals in the definition of ~,,,y,. Combining this with (3.8) completes the proof.

Using (3.3) instead

of

[]

(3.1)

in the above proof yields the corresponding formula

for the derivative: Theorem

3.3 Let x , y E ~ ( H ) .

If f : R --+ C is an odd function for which

rn] E 9.1s(R), ~hen

f'(~).y = ~.,x(~).

(a.9)

The class of functions ~is(R) has a rather abstract definition. In [4, section 7] Birman and Solomyak identified some concrete classes of functions f for which rn] is contained in 92s(R). These properly include functions whose first derivative has an integrable Fourier transform and functions whose first derivative is in both Lipe for some e > 0 and LP(R) for some 1 < p < oo. An additional concrete class of functions with this property is given in section 4 below. R e m a r k 3.4 Each partial isometry v in ~ ( H ) , or more generally each tripotent v in a JB*-triple A, determines the Peirce decomposition A = A2(v)| AI(v)~Ao(v).


473

T h e associated Peirce projections Pk(v) : A --~ Ak(v), k = O, 1,2, are contractive. In the fl3(H) case the Peirce projections have the form

P2(v)z = vv* zv*v,

Pl(V)Z = vv" z(1 - v'v) + (1 - vv*)zv*v, (3.10)

Po(v)z = (1 - vv')z(1 - v ' v ) . An orthogonal family {vj}~ of tripotents determines the decomposition A =

O Pj,k A, O