The Hellinger square-integrability of matrix-valued measures with respect to a non-negative hermitian measure

A R K I V F O R M A T E M A T I K B a n d 7 n r 21 1.67 05 1 Communicated 8 March 1967 by H. CI~AMEI~

The Hellinger square-integrability o f matrix-valued measures with respect to a non-negative hermitian measure

B y H A R m SALES~

Introduction

L e t B be a a-algebra of subsets of a given space ~ , a n d let F be a fixed non- negative h e r m i t i a n measure on B. F o r m a t r i x - v a l u e d measures M a n d N t h e Hellinger integral .~a(dMdN*/dF) (* = c o n j u g a t e ) is defined in such a w a y t h a t t h e space of all m a t r i x - v a l u e d measures M for which ~a(dMdM*/dF) exist be- comes a Hilbert space u n d e r the inner p r o d u c t v.~a (dMdN*/dF) ('c = t r a c e ) . I t will follow t h a t ~a(dMdM*/dF) exists iff there exists a B-measurable m a t r i x - v a l u e d f u n c t i o n W on ~ such t h a t W E L2.r [9, p. 295], a n d for each B E B, M ( B ) =

~BtlSdF. These generalize the corresponding results [3, pp. 258-61] & [8, pp.

1414-18] concerning t h e Hellinger integrals ~a (dvd~/dtt), where r a n d 7 are com- plex-valued measures a n d # is a n o n - v e g a t i v e real-valued measure on B.

F o r a n y m a t r i x 6[ we write G- for t h e generalized inverse of G [7, p. 407].

If tt is a a-finite n o n - n e g a t i v e real-valued measure on B with respect to (w.r.t.) which F is absolutely continuous (a.c.), t h e n it is easy to show t h a t (dF/dlt)- is a B-measurable m a t r i x - v a l u e d function on g2.

Lemma 1. Let (i) M and N be matrix-valued measures on B.

(ii) # and v be q-/inite non-negative real-valued measures on B w.r.t, which M, N and F are a.c. 1 Then

(a) f ~ (dM/d#) ( d F / d # ) - (dN/d#)* d~ exists i//

f (dM/dv) (dF/dv)- (dN/dv)* dv exists.

(b) I/ these integrals exist, they are equal.

Proo/. (a) L e t V = # + v . If ~ a ( d M / d # ) ( d F / d # ) - ( d N / d # ) * d/~ exists, t h e n f r o m t h e relations

1 Each matrix-valued measure is a.c.w.r.t, the sum of the total variation measures of its components. Hence such a # exists.

It. SALEHI,

HeUinger square-integrability of matrix-valued measures

f (dM/dp)

(dF/d#)-

(dN/d#)* d# f~

(dM/d#) ( d F / d ~ ) ( d N / d # ) * ^I

(d/~/dy) d~

f (dM/d~) (dF/dT)- (dN/dT)* d T

(1)

f~ (dM/dT) (dF/dT)- (dN/dT)* d 7

follows that exists.

Conversely if

fa (dM/dT) (dF/dT)- (dN/dT)* d7

exists, again from (1) we infer that

f (dM/d#) (dF/d#)- (dN/d/x)* d#

^exists.

Similar argument can be used to show that

f~ (dM/dv) (dF/dv)- (dN/dv)* dv

exists iff

fa(dM/d~) (dF/dT)- (dN/dT)* d 7

^exists.

Hence (a) is proved.

(b) From the argument used in the proof of (a) we infer (b). (Q.E.D.) Thus the following definition makes sense.

Definition 1.

Let

M, N, F

and # be as in the previous lemma. Then

(a)

we say that

(M, ~)

is Hellinger integrable w.r.t. F i/ S~ (dM/d/~) (dF/d/~)- (dN/d#)* d/~ exists.

We write

f dM dN* ; dF j~ (dM/d/x) (dF/d#) (dN/d#)* d/x.

(b) H2. F

is the class o/all matrix-valued measures M on B/or which Sa (dMdM*/dF) exist.

I t is easy to see that

(1) M, N E H 2 . v * (M, N) is Hellinger integrable w.r.t. F, MEH2, F and A is a m a t r i x ~ A M E H 2 , v ,

M,N E H 2 , F ~ M + N E H2, r.

B y (1)

Sa(dMdN*/dF)

exists for M, N E H2. F. This matrix-valued integral be- haves like an inner product. I t i~ therefore convenient to write

ARKIV FOR MATEMATIKo Bd 7 nr 21 (M, N)r = f ] d M dN*

dF J~

We define the ordinary inner product for H2. r by ((M, N))F = T(M, N)F.

Thus from (1) we immediately get:

Lemma 2. H2,F

is an inner product space under

((., .))r,

where/or M and

NE He,r ((M, N))r = ~(M, N)r.

Let Le.F be the class of all matrix-valued functions r on ~ for which ~ n ~ d F ~ * exist (A detailed discussion of integrals ~nCdFW* and j ' ~ r are given in [6]

and [9]). It is known [9, p. 295] that L~,F is a Hilbert space under the inner product.

((q~ W))F = ~(r W)r = ~ | ~ dFW*.

J~

The following lemma is needed to establish an isomorphism between L2, F and

1:[2, F 9

Lemma 3.

Let (i) ~ and

W EL2, r (ii)

_For each BE B

M ( B ) = ~ * d F

and

N ( B ) = f B W d F . Then (M, N) is Hellinger-integrable w.r.t. F and

(M, N)F = ( ~ , W)~.

Proo/.

Let # be any a-finite non-negative measure w.r.t, which F is a.c. Then for each B e B, M(B) = ~Br d/x and N(B)

= fB W(dF/d#) d#.

Hence

(dM/d/z) = r (dN/d/x) = W(dF/d#).

Therefore

f dM

dF dN* - f a (dM/d#)

(dF/d#)

(dN/d~)* d#

= f c~(dF/d#) (dF/dlu) (dF/d#) ~r~* d~

= f O(dF/d~) ~* d# = (~, ~)~.

⁽¹⁾ :Since for r and t~'tEL2,F, (CI~,~)F exists, from (1) it follows that (M,N)r is

Hellinger integrable w . r . t . F . Moreover (M, N)r= (~, W)F. (Q.E.D.)

H. SALEHI, HeUinger square-integrability of matrix-valued measures

If W E L2. F, Then M~, will denote the matrix-valued measure in H2. y such t h a t for each B E B, M ~ ( B ) = S, WdF. Hence the following definition makes sense.

Definition 2. Let the trans/ormation T be de/ined ^{o n} L2,y into H2,y ^{a s} /oUows:

TW = M~.

The important properties of T are given in the following theorem:

Theorem 1. (a) T is a linear operator on L2. r into H2,F, i.e., i/ A and B are matrices and r and t~ E L2.F, then

T ( A ~ + BW) = A T ~ + BTW.

(b) T is an isometry o n L2, F into H2.y. I n /act (T~, TW)F = ( ~ , W)F.

(c) T is onto H2.F, i.e., /or each MEH2,F, there exists a tIYEL2.F such that M = TW. I n /act we can take tit to be (dM/d/~) (dF/d[~)-, where # is any a-/inite non-negative real-valued measure w.r.t. M and F' are a.c.

Proo/. (a) and (b) follow from L e m m a 3 and Definition 2.

(c) L e t ME H2. ~. I f # is a n y (~-finite non-negative real-valued measure on B w.r.t, which M and F are a.c., then

(M, N)F = f ~ (dM/d#) (dF/d/~)- (dM/d/~)* d#

= f a [(dM/d/~)(dF/d/~)-] (dF/d/~)[(dM/d#)(dF/d/~)-]* d/~,

where the first equality follows from the definition of (M, N)F and the second one is a consequence of (dF/d[~)-(dF/dl~ ) (dF/dl~) = (dF/d/~)-. Hence (dM/d/~)(dF/dp)- is in L2. r. Let N(B) = ~B(dM/dl~) (dF/d/~)- dF. Then (M, M)F = (N, N)F a n d (M, N)F =

(N,M)F. Hence ( N - M , N - M ) F - 0 , i.e., N and M as elements of H2.F are equal.

B y Definition 2, T(dM/d#) (dF/d/~)- = ST. Therefore T(dM/d/~) (dF/d/~)- = M.

(Q.E.D.)

We immediately obtain the following result.

Theorem 2. (a) H2. r is a Hilbert space under the inner-product ( ( ' , ' ) ) F . (b) M E H2,F ill there exists a B-measurable matrix-valued/unction W on ~ such that • E Le.F and /or each B E B,

= I ~ dF.

M(B) JB

Moreover i/ # is a (y-/inite non-negative real-valued measure w.r.t, which M and F are a.c., then ( d M / d / ~ ) ( d F / d # ) - ( d F / d # ) = (dM/d/~) a.e. /~.

ARKIV FOR MAT]~MATIK. B d 7 n r 21

Proo/.

(a) (b). (a) and the first p a r t of (b) are immediate consequences of Theorem 1. F o r the second p a r t of (b) we have

(dM/d#)

(dF/d/~)-

(dF/d#)= tY(dF/d#) (dF/d#)-

(dF/d/~)

= W(gF/diu ) = (dM/d/~) a.e. /~,

where the first and the third equalities are consequences of M ( B ) = S B t F d F and the second one is a consequence of (dF/d/~)=

(dF/d#)(dF/d/~)-(gF/d/~)

a.e. /~.

(Q.E.D.)

Remark.

The significance of the Hellinger integrals S~

(drdT-/d#),

where r and 7 are complex-valued measures and/~ is a non-negative real-valued measure, in uni- variate utochastic processes has bee pointed out b y H. Cram~r [1, p. 487] and U. Grenander [2, p. 207]. Our Hellinger integrals play an i m p o r t a n t role in q-variate stochastic processes. I n particular, t h e y give rise to an extension of P. Masani's work on q-variate full-rank minimal processes [5, pp. 145-150] which in turn is a generalization of a well-known result of A. N. Kolmogorov on uni- variate minimal sequences [4, Thm. 24]. These and other results will be published separately.

A C K N O W L E D G E M E N T

This research was partially supported by National Science F o u n d a t i o n GP-7535.

Michigan State University, East Lansing, Michigan (U.S.A.)

R E F E R E N C E S

1. CRA~R, H., Mathematical Methods of Statistics, Princeton University Press, New Jersy, 1961.

2. GRENANDER, U., and SZEG6, G., Toeplitz Forms and Their Applications, University of Cali- fornia Press, California, 1958.

3. HOBSON, E. W., On Hellinger's integrals, Proc. London Math. Soc. 2, No. 18, 249-265 (1919).

4. KOLMOGOROV, A. N., Stationary sequences in Hilbert space, Bull. Math. Univ. Moscow 2, No. 6, 1941.

5. MASANI, P., The prediction t h e o r y of multivariate stochastic processes, I I I , Acta Math. 104, 142-162 (1960).

6. MASANI, P., Recent Trends in Multivariate Prediction Theory, MRC Technical S u m m a r y Report No. 637, J a n . 1966, Math. Res. Center, Univ. of Wisc.

7. PENROSE, R. A., A generalized inverse for matrices, Proc. Camb. Phil. Soc. 51, 406-413 (1955).

8. RADON, J., Theorie und anwendungen der absolut additiven mengenfunktionen, Sitzsber.

Akad. Wiss. Wien. 122, 1295-1438 (1913).

9. ROSENBERG, M., The square-integrability of matrix-valued functions with respect to a non- negative hermitian measure, Duke Math. J. 31, 291-298 (1946).

Tryckt den 29 december 1967

Uppsala 1967. Almqvist & Wiksells Boktryckeri AB