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Operational Calculus for Two Commuting Closed Operators

J. T. MARTI (EIR, Wiirenlingen)

87

I. Introduction

Functions of an endomorphism T of a complex Banach space X can be defined through a proper homomorphic mapping qJ of an algebra H into the Banach algebra B(2E) of all endomorphism of X. The elements of H are locally holomorphic complex functions f (2) defined on a neighborhood of the spectrum a(T) of T. If q) is con- tinuous and the image of 2 under q) is T, then 9 is unique and is represented by a Cauchy integral: q~(f)=(Zni)-'Sr f (2 ) (2 -T ) - ld2 , where F is a suitably chosen curve in the complex plane, q~(f) is a function of T, and the construction of functions of T by this procedure is known as operational calculus for bounded linear operators. The Cauchy formula has been used by DUNFORD [2, 3, 4] and by TAYLOR [10] to define functions of an endomorphism of ~. Such a function could be defined also by a power series expansion in T, valid in a neighborhood of o(T), and the result is equivalent to the result obtainable with Cauchy's formula. However, if Tis unbounded and closed, a generalization is possible only by use of the integral formula (TAYLOR [11], HILLE and PHILLIPS [7]). Another extension of the theory is the generalization to functions of more than one variable, which goes straightforward only in the case of commuting endomorphisms of X. Such functions have been introduced by SCHWARTZ [8] and WAELBROEK [13], where uniqueness is proven in the last cited work.

The theory presented in this note deals with functions of two (not necessarily bounded) commuting closed linear operators T 1 and T 2 on 2E to itself. Again, the functions are defined through a proper homomorphism q~ of an algebra H into B(~). Here H consists of locally holomorphic complex functions f(21, 22) whose domain V is a neighborhood of oe(7"1) x %(T2) , O'e(Tj) being the extended spectrum of Tj. This implies that V is unbounded, if T 1 or T2 is unbounded. In principle, the main theorem describes the following facts: (2 - Tj.)- 1 for j= 1, 2 and for each 2 in the representation given by formula (1).

Provided that q~ transforms (2-2 j ) -1 into the resolvent set of Tj,~ is unique and has

The operational calculus for closed linear operators plays an important role in the theory of Banach algebras [12, 14], in the spectral theory [4], the theory of semi- groups [7] and in quantum theory [1 ]. As an application we shall give a representation for polynomials in two endomorphisms of . Furthermore, we derive formulas for the resolvents of the sum and the product of commuting closed linear transformations and show that the extended spectrum of the sum (product) is contained in the sum (product) of the extended spectra of the single transformations.

88 J .T . MART1

2. The operational calculus

Let V 1 and V2 be Cauchy domains, open subsets of the extended complex plane C with a finite positive number of components whose boundaries are nonempty and are composed of a finite positive number of simple closed rectifiable curves. Let V= V~ V2 be the Cartesian product of V 1 and V2 in C 2. We denote by Xx the charac- teristic function of a set X. Let H(V) then be the complex algebra of all functions f :C2-oC which are locally holomorphic in V and vanish in the complement of V with respect to C 2. In H (V) we use the ordinary definition of arithmetic operations. Xv is the unit element of H(V).

Now let U be a Cauchy domain and X a closed subset of U. Then there exists a Cauchy domain U', such that Xc U' and U' c U [11, Theorem 3.3]. The union F of all boundaries of U', with the usual positive orientation, we call an oriented envelope of X with respect to U.

Next, let 3s be a complex Banach space, s (3s the class of all closed linear transfor- mations with domain and range in 3s and B(3s the complex Banach algebra of all endomorphisms of 3s with identity I. Let T be an element of s (3s with domain ~3 (T), spectrum a(T) and resolvent set a(T). The extended spectrum ae(T) is the set of all singular points of the resolvent R(2, T) of T in C. By ~(3Q we denote the class of all pairs { T x, T 2 } of operators of s (3s which satisfy ~ ( T 1) c ~3 ( T 2), T 2 [~) ( T 1)] c ~(T1) and commute on the set {xlxE~(T1), Tlx~7~(Tz) }. If T 2 is bounded on 3s this definition is equivalent to that of M. H. STONE [9, Definition 3]. Then we have

LEMMA 1. Let ~ (7"1) c ~ (T2) and T 2 [~3 (T1) ] c ~ (7"1). A necessary and sufficient condition that R(2x, 7'1) R(22, T2)=R(22, T2) R(21, T1) for each pair {2x, 22}e0(T,) x 0(T2) is that {T 1, T2}e~(3s

Proof. If {T 1, T2}eff(3s ) we have {x[xeTs Tlxe~(Tz)}={x[x~7~(T1), (211-T1)xe 33( T2) )= R(21, T1)[3)( T2)]= R(21, 7"1)R(22, T2)(3s since the R(2s, Ts) are one-to-one transformations of 3s onto ~)(Tj). For each xe3s

R(~, r~)R(~, W,)[(a~ ~- T~)(2~ I - r~) - (~ ~- W~)(~1 ~- W,)] x R(2,, T,)R(22, rz)x = [R(2,, T,) R(22, Tz) - n(2z, T2)R (2,, Tx)] x.

Hence a sufficient condition for the right-hand side to vanish is that T 1 and T z commute, i.e. { 7"1, T2} e ~(3s On the other hand, if the right-hand side is zero, again since the resolvents are one-to-one, we have (7"1 Tz - Tz T1) R(21, T1) R(22, T2)x= O. Thus { T 1, Tz} ~(3s and this proves the necessary condition, q.e.d.

Finally, we denote by (5(3s V) the class of all elements of ~(3s for which ae(T1) tre(T2) c V. Based on the preceding definitions we have

Operational Calculus for Two Commuting Operators 89

THEOREM 2. For each pair { 7"1, T 2 } ~ ffJ (3~, V) there exists a proper homomorphism of H (V) into B(X) such that

(i) ~[(2 -2 j ) - 'Xv(2 , ,22) ]=R(2 , T~), 26Vj, j= l ,2 .

(ii) I f a sequence fn in H (V) converges pointwise to fe H (V), the convergence being uniform on each compact subset of V, then this implies lim,[r@(f,)-#(f)]J =0.

(iii) 9 is unique and is defined by

e( f ) = f(oo, o~) ~+ 2~ f(,h, oo) g(,h, ra) d2, FI ,f +2~/ f(oo,2~)g(,h, r~)d,h (1)

F2

, - - , l f f + ~ f(21, 22) R(2,, T,)R(22, T2) d2 , d22, FI F2

where F j, j= l, 2 is an oriented envelope of ae( Tj) with respect to Vj, containing 2= oo in its interior if Vj is unbounded.

Proof. Since the integrands of (l) are locally holomorphic in Vc~ [0(T,)x 0(T2)] the integrals exist and are independent of the choice of F 1 and F 2. Clearly q~, defined by (1), is linear. In order to demonstrate that it is a homomorphism we taker, g~ H(V) and for j= 1,2 two oriented envelopes Fj and Fj. of ae(Ti) with respect to Vj such that the open set bounded by Fj contains F~.. Then by (1) with c=(2~i) -1,

8

q~(f) q~(g) - ~0(f g) = ~, Ak, (2) k=l

where

A1 = c ~ I f (11, Go)g(oo, oo) + f (o% oo)g(21, oo) - (f g) (11, O(3)] R(21, 7",) d J, 1 Fl

A 2 = c f [ f (~, 12) g (~, ~) + f (o% oo) g (~, 12) - ( f g) (~, 12) ] R (,~, T~) d22 Fz

A3 = c 2 ~ f f f (11, oo)g(pl, o0) R(21, 7"1) R(#,, T,)d21 d/~, F1 FI"

A4=c2 f f f (~176 g(~176 T2) R(p2, T2) d22dp2 F2 f2'

=c2 (; [f(~,22) g(2 I, ~)+ f(2,, ~)g(00,22)+ f(~, ~)g(2,,22) As F1 F2

+ f (2,, i2) g(oo, o0) -- (f g) (2,, 12)] R(2,, T,) R(22, T2) d2, d22

90 J.T. MARTI

A6=c3f f fry (,,, ~)g(X,, ).2)+ f ('~'1, ~'2)g(#l, o0)] r l F1' F2

x R(2,, T1)R(//1, T1)R().2, T2)d2, d#, d22

Ft F2 F2'

R(~,, rl) R(~2, r~) R(U~, r~) d~, d~ d~

.,~8=c4f f f f f(~l,~2) g(~ll,~12)R(~l, T1)R(~2, T2)R(~/ , , T , ) FI F2 FI' F2'

R(H 2, r2) d21 d22 d/~l d/~2.

In some terms we have changed the order of integration and used the commutativity of the resolvents (Lemma 1). By [11, p. 196] we have for j= 1, 2 and any hell(V) as a result of the method of residues

and

f h (#j, ") c p j -2 j dpj = h(2j, " ) - h(oo, "), 2jEFj. r j

c I -h% ) a.~ = - h(oo,-), ~r j .

Fj'

Using this and the resolvent equation (#1-21)R(21, T1)R(pl, T~)=R(2 x, Tx)- g(#1, T1) we get

A~ =c ~ f R(x,, 7"1)f(21, oo)d21 f g(pl'~-! - '~1 FI FI'

+ c 2 R( . . r0 g(~, ~)dp, i~ _~, FI' FI

=- -A 1

and similarly A4= -A2. In the same manner we obtain

&=c3f f f [f (2'' 22) g(l~l' 22) - f (2~' ~) g(#~' 22) - f (2~' 22) g(l~'' ~176 FI FI' F2

R(,h, V~) R0,. V~) R(.h, V~) d,~ d~l d~,

Operational Calculus for Two Commuting Operators 91

hence

h6+as=c3f f ff(21,"~2) g(#l,22)R(21, T1)R(//1, T1) R(22, T2) d21 dpl d22 FI FI' F2

=c2 S f [(Sg)(~,,a~)-S(oo, a~)g(a,,a2)-s(a,,X2)g(oo, x2)] /'1 F2

x R(,h, 7"3 R(22, 7"2)d2, d,h.

Since analogously

f f r- s (oo, 22)g(2,, oo)- s(2,, oo) g(oo, 22)- S(oo, oo) A7=c 2 Fl F2

g(2,, 22) - f (~,,, 22) g(oo, oo)

+ f (00, 22) g (2,, 22) + f (2,, 22) g (00, 22)] R (2,, T,) R (22, T2) dXl d22,

we have A7+As=-A6-As , so that the right-hand side of (2) vanishes and 9 (fg) = q~(f) ~(g).

Next, to prove (i) we write down two well-known relations from the operational calculus in one variable [7, Theorem 5.11.2]

c f R(2j, T1) d2j = [1 - Zvj(OO)] I (3) Fj

and [11, Theorem 7.4] (after deforming the path of integration)

c .f (2 - 2i)-1 R (2j, Tj) d2j = n (2, Ti), 2 ~ Vj. (4) Fj

Using equations (l), (3) and (4) we get

9 [(2 - 2 , ) - ' Zv(2,, 22)] = Zv2

92 J.T. MARTI

_c2 k , l= l

Now we take a pair {#1, ]/2} of finite POints of C2- IT. ForgeH(V)wehave by (1)

f --j g (21 ,22) [ (21-px) - ' (22- /~2) - ' I - (21 - -p , ) - 'R (22 , T2) - t~(g) = c 2

Ft F2 - (22 - /~2)- ' g(2, , T,) + g(2, , Ti) R(22, T2)] d2, d22.

Letf. ~ H (V) converge uniformly tof~ H (V) on F 1 F2. Then lira, [[ 9 (f,) - 9 (f)[] = 0, since FI F2 is compact and the expression in the square brackets is bounded on F x F2. This proves the continuity property (ii).

In order to prove the uniqueness of (1) we take a homomorphism of H (V) into B(X) which satisfies properties (i) and (ii). Then we show that 9 has the desired representation (1). Due to (ii) we have in the uniform operator topology q~(f )=l im.~(f . ) , if there exists a sequence f . in H(V) converging pointwise and uniformly to f~H(V) on each compact subset X of V. But by Cauchy's formula we have on X= X 1 X 2 :

f(2t, 22)=f(oo, oo)+ c f(~l, )dkq+ c p/__~;_d, 2 ,111 - -

FI" F2"

(p i 2- ) (p2--~_ 22) dP, dl~2, FI' 1"2'

where F~ are oriented envelopes of Xjw ae(7"i) with respect to V 3 containing 2= oo in its interior if Vj is unbounded. By Vj. we denote the Cauchy domain enclosed by F~. Since Xj w ae(Tj) is closed and the boundary Fj of V i is compact, the distance 6j between Xjwae(Tj) and Fj is positive. Fj. may be chosen such that the greatest distance between Fj and the points of F~. is less than 89 i, 1/Mj}, where Mj=sup{l[R(2j, Tj)III2fiPj-Vj}. Hence there exist suitably chosen sequences of points I~jl ..... i~j,=l~jo on Fj and vjl ..... vj.= Vjo on Fj, such that I~jk-~j, , - l l - - '0 uniformly with respect to k for n~ and Ivjk--~jkl 0 an m such that on Xfor n>~m

~ ~lk -- /Al'k- I f (2 , ,2z ) - f (~ ,~) -c p-~__).l f( l~'k'~)

k=l

~ JLL2 ! -- P2,1-1 - c

/=1

(/q * -/~1' k-') (/~2 t --/~2"- t) f #2,)

Operational Calculus for Two Commuting Operators 93

Unfortunately, this approximation forfdoes not belong to H (V), so that we have to move the singularities out of V. But [(vjk--pj,)/(Dk--2j)l 1 mj

n.t

(~j~ 2j) -1 / i=0

where Lj is the length of F~ and N= sup {l f(2,, 22)1 2,eFj, ~2~r~). Thus

I f (21, J '2) - - fn ('~1, '~2) I < 3 e + 82/N on X, where n>~max{m, m 1, m2} and

x (V,k_21),+ ~ +C ~) f (~, P2,) (v2t -- P2t)' i=0 i=0

+ ~ Z (.,~ - ~1.k-,)(~2, - .~ , , - J f ( . ,~ . .~ , ) k , l= ,

n

(v,~ - a , ) '+ ' (~, - a21 j+ ' J i , j=O

Clearly f~ is a sequence with the required property and by (i) we have in the uniform operator topology

= (#,k -- /~, ,k - ,) f (/~, k, o0) q~(f) f(oo, oo) I+ l im, Ck 1

~ (V,k--P,k)'R(V,k,T,y +' +c ~ (t~2~-1~2,,_,)f(oo, p21) i=0 1=1

~ (v2t-1~2t)iR(v2t, T2)i+'+c 2 ~ (P,k--P,,k-,)(#2,--P2,,-*) i=0 k , l= l

xf (p ,k , P2,,) ~ (V,k--#tkf(V2t--P2,) j i+'R(p2,,T2)/+'}. i , j=O

From the resolvent equation we easily get

(Vjk -- lajk)iR(vjk, Tj) i+1 = R(pjk, Tj) { I - [(Vjk -- I~ik) R(VjR, Ti)] "+1} I=0

94 J T. MARTI

so that, since II(vjk--~jk) R(Vjk, rj)]l < 89

~ ( f ) = f ( oo , oo ) I + c f f ( ). , , oo ) R ( ). , , T, ) d ~ , + c f f ( oo , ). 2 ) R ( ). 2 , T2 ) d ~ 2 El ' F2'

+c 2 f f f ( ) . " ) .2 )R( ) . "T ' )R( ) .2 'T2)d2 'd22 FI ' F2"

and the proof of Theorem 2 is complete.

3. Polynomials, resolvents and spectra of sums and products of operators

In view of an application to polynomials in bounded linear operators we have the following

COROLLARY 3. Let Vj be bounded and let { 7"1, T 2 } E (~ (~, V). Then for j = 1, 2 we have ~[).jZv(21, ).2)] = Tj.

Proof. Since Tj is bounded it immediately follows from (1) and (3) that

1 t ~j R ().j, Tj) d).j [). j x . ( ) . , , ) .2 ) ] = 2%--i i r

Fj

_ 1 ~[I+TjR().j, Tj)]d2j=Tj. q.e.d. 2hi 3

rj

For the sum of T 1 and T2 we obtain

THEOREM 4. Let {T1, T2}~ ~(X, V). Then for each Xq~ V~ + 1/2 we have

q~ [(). - )., - ).2)-' Zv(2,, ).2)] = R()., T, + T2).

Proof. Clearly ().-21-).2)-1Xv(21,).2)~H(V) if there is a ).~VI+ V2. Applying the method of residues we get by (1)

1 f R ().,, T,) R (). - ).,, T2) d)., [(2 - 2, - 22)- ' Xv ().,,).2)] = 2 n--~ FI

(5)

If V1 is unbounded, then, according to our assumptions, V2 must be bounded and vice versa. On ~ we then have

Operational Calculus for Two Commuting Operators 95

(it I - 7"1 - w2) f R(it,, T1) R( i t - it,, T2) xdit , FI

= [itx I - T, + (it - it,) I - 7"2] f R(itl, T1) R(it - it1, T2) x dit, FI

- - f (it- itl, dit, + 5 (itl, air, Ft FL

= - Xv,(OO) f R(it - it,, T2) x dit I q- 2~ i[1 -- Xv,(OO)] X

C1

T2) xdp+2n i [ I - )~v,(OO)] x CI

=2nix ,

where we have used the fact that T 1 and T 2 commute with the integral, C1 is a posi- tively oriented circle of sufficiently large radius around the origin, and/~ has been substituted for 2-21. Hence the convolution (5) equals R(it, T 1 + T2), q.e.d.

COROLLARY 5. Let T 1 or T 2 be bounded and {7"1, T2}ff~(~ ). Then

ae( 7"1 + W2) c ae( 7"1) + ae( r2).

Proof. If a,(T1) + a,(T2) = C the statement is trivial. Otherwise we have for each complex 2r a neighborhood V (in the sense of Section 2) of ae(T1)xa, (T2) such that ( i t - i t l - i t2) - lZv( i t , , it2)~H(V) and the resolvent R(it, 7"1 + 7"2) is given by (5). But since i t - i t l~a,(T2), the integrand in (5) is bounded on F 1 and, since F 1 is compact, R(it, T 1 + T2) is also bounded. Hence i t~a,(T 1 + T2). This implies a,( T 1 + T2) ~ ae( T,) + a~( T2), q.e.d.

In a manner similar to that used in Theorem 4 we obtain for the product of T 1 and T2:

THEOREM 6. Let { T1, T2}~(~(X , V). Then for each it4g Vl"V2 we have

[(~ - ~, ~)-' z . (~i, ~) ] =

- - 2Toil R ( i t l , T,) R ( i t / i t , , 7"2) ~- = R (2, T, T2). (6)

FI

The proof of this theorem parallels that of the preceding theorem. Here we have to show that on

96 J.T. MARTI

f d,,lq (2 I - T, T2) R(2,, T1) R(2/2,, T2)x 2--7 Fj

= 21-21 T2-- I - T 2 (21 I -T1)+ I -T , )

El d21

R (2,, T1) R (~./2,, T2) x - - ;h

~ fd/~l ; d~l = R(J.t,T,)xd2 , - 21 x+2 R(2/2,,T2) x 22

FI F1 F1

= 2 n i[1 -- Zv, (m)] x...