On the continuity of a reduction-amplification operator in quantum mechanics

On the continuity of a reduction-amplification operator in

quantum mechanics

Citation for published version (APA):

Eijndhoven, van, S. J. L., & Graaf, de, J. (1984). On the continuity of a reduction-amplification operator in quantum mechanics. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8407). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1984

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 84-07 May 1984 ON THE CONTINUITY OF A REDUCTION-AMPLIFICATION OPERATOR IN QUANTUM MECHANICS by

S.J.L. van Eijndhoven and J. de Graaf

Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands

ON THE CONTINUITY OF

A REDUCTION-AMPLIFICATION OPERATOR IN QUANTUM MECHANICS

by

S.J.L. van Eijndhoven and J. de Graaf

Abstract

In the Banach algebra B(H) of bounded linear operators on a separable Hilbert space Owe investigate the continuity of the linear mapping

<nolAlm.o> Inn>

<ron

I

with respect to the uniform, the ultra-strong, the strong and the weak to-pologies on B(H).

2

-STATEMENT OF THE PROBLEM

Let

H

be a Hilbert space. Let

B(H)

denote the Banach algebra of bounded

*

linear operators on

H.

Informally the operator P :

B(H)

+

B(H)

is defined

c by means of Dirac brackets in the following way

*

PcA

=

I_

<nOIAlmO>lnn>

<ron

I .

nmn

Here the operator A is supposed to have the matrix representation

A

=

J _

<nIiIAItmD>Inn>

<mIDI

nnmm

The problems posed by W.M. de Muynck, cf. [M], are the following:

*

(i) Show that for each A

E

8(H) the operator

P

A is well defined and

c belongs to

B(H).

*

(ii) Investigate the continuity of the mapping

P :

B(H) + B(H) in the c

uniform, the strong and the weak topologies of B (H) •

MATHEMATICAL FORMULATION AND RESULTS

<X> Let

X

be a separable Hilbert space with a fixed orthonormal basis (v) O' _{n n=}

00 Let

H

=

X

e

X

denote the two-fold tensor product of

X.

Then (v

e

v_) - 0

n n n,n= is an orthonormal basis in

H.

In X define the operator P by P f = (f ,v )Xv •

nm nm n m

In

H

the operator P

e

I can be written nm (P

e

I)F = nm 00

I

(F,v

e

v-)H(v ® v~) .

n=O

n n m 11

3

-Next, let A be a bounded linear operator on H, i.e., A € B(H), and define.

co

the matrix ( a ) _{om n,m=} 0 by

OUr first problem is to give a mathematical meaning to the sequence

L

I

n=O m=O a

om (P om ® I) •

Let Po denote the projection in

X

defined by POf = (f,vO)XvO' The projection operator I ® Po on

H

can be written

QO

t

_{(F'Vn ® vO)H(vn ® vO' .}

n=O

For any operator A € B(H) the operator (I 0 PO)A(1 0 PO) ~ B(H) •

-We have II (I ~ PO)A(1 ® PO) liB (H) ~ ilAII_BCH) •

Further (I ~ PO)A(1 ® PO' maps H = X ® X into X ® <va> and also X ® <va> into X @ <va>. It is c:ear now that (I ® PO}A(I @ PO) can be regarded as a mapping from

X

into

X.

This so-called ~eductiod will be denoted by AO'

*

Finally, we are in a position to define the operator P

c

p* B (H) -+ B (H) c

Calculation of P*A(V ® v ) shows that p* is indeed the desired operator

c p q c

as mentioned at the very beginning of this notice.

*

Remark. P (Q ® I)

=

Q ® I for all Q E B(X) .

4

-THEOREM 1

*

P is a bounded linear operator on B(9) and c

*

Hence P is continuous in the uniform topology of B(9). In other words c

*

*

Pc E B(B(H». Finally IIP cIlB(B(H»

=

1. PROOF

*

IIP_CAII_B(9) = HAO 0 1IIB(H) == IIAOIlB(X) :: II (I 0 PO)A(1 0 PO) IIB(H) :;i IIAIiB(H)'

For the special choice A == Q 0 PO' Q E B(X), we have

For the used properties of Hilbert norms of tensor products of Hibert spaces see Weidmann [W].

LEMMA 2

Consider the linear mapping

r :

B(X) + B(O) : Q ~ Q Q I.

r

is the so-called 'amplification map'. See Dixmier [D].

(i)

r

is continuous with respect to the uniform topologies of B(X) and

B(O) .

(ii)

r

is continuous with respect to the ultra-strong (= strongest, [N]) topologies of B(X} and B{O) .

(iii)

r

is sequentially continuous with respect to the strong topologies of B(X) and B(9).

(iv~

r

is sequentially continuous with respect to the weak topologies of B(X) and B(O).

5

-PROOF:

(i) Trivial because IIQIIB(X) ::: IIQ ® IIIB(H)

""

(ii) Take a sequence (Fk}:=O c: H such that

I

II (Q S

I)Fkll~

< "". k=O (iii) (iv) 00

I

II(Q ®

I)Fk"~

k:::O 00 v. with fk 1.. • 1. E: X. Then 00 00 =

I

L

IIQfk

II~

k=O k=i i

Let (Q) _{n n=} 1 c:B(X). Suppose Q _n + 0 stronNly, i.e. for all ';:I

0)

f E: X, UQnfllX + O. Further let F =

L

fk S v_k € H. We must show that k=O

II(Qn @ I)FII

H + O.

From the Banch-Steinhaus theorem it follows that

is a bounded sequence. From this and the strong convergence of Q n it follows that II (Qn @ I) FII H + O.

0 )

Let (Q) 1 c: B(X}. Suppose Q + 0 weakly, i.e. for all

n n= n _{00 CIO}

f,$

€

xl

(Q

f,$)xl

+ O. Let F:::

I

n k=O

We must show that

I

«Qn ® I)FI~)H) + O. This follows from 00

6

-OC>

and the uniform boundedness of the sequence (Qn)n=l' in a way

similar to (iii). [J

THEOREM 3

The mapping

P; :

E(H) + E(H) :

A~

AO ® I is

(i) continuous with respect to the uniform topology on E(H},

(ii) continuous with respect to the ultra-strong topology on E(H),

(iii) sequentially continuous with respect to the strong topology on E(H),

and

(iv) sequentially continuous with respect to the weak topology on B(H) . PROOF:

*

We write

P

as a composition of linear mappings c

p* B(H) + B(H) + B(X) + E(H) c

The desired continuity of the first arrow follows from Naimark [NJ, Ch. vii. See also [DJ. The desired continuity of the second arrow follows because

x

+

X

® <va> is a unitary bijection. Finally, the desired continuity of the third arrow follows from Lemma 2. [J

Next we want to show that the property of sequential continuity in Theorem 3, (iii) and (iv) cannot be replaced by continuity.

LEMMA 4

The amplification map

r

E(X) + E(H) E

*

E 0 I is neither strongly nor

7

-PROOF:

(i) Consider the set of operators

""

Here F =

r

fk 0} v

k is taken fixed. We have

l:

Ilfk"; < "" and we

k~ k~

take (fk)~=O ~

X

such that the span of the fk is dense in

X.

Now suppose that there exists a finite sequence (~l""'~p) ~ X and

o

>

a

such that the set

is contained in SF" By taking Q = ~IT € S~ i~ where

rr

is the

. .1'···' P

projection operator onto the orthocomplement of span <~1"""~~ and ~ >

a

sufficiently large, we get a contradiction.

(ii) Consider the set of operators

co co

where F

=

r

f

k @ vk and G

=

r

gk @ vk are taken fixed. We choose

k=O -1 ~=O

fk ::: gk::: (k+l) V

k• Now suppose that there exist two finite sequen-ces (~l""'~p)' (Wl,···,W_p) ~

X

and 0 >

a

such that the set

R h. h. < =

{Q

j

I

(Q4>. ,

W . )

I

< <5, 1 :ii i ;ii

p}

't'1""''t' ,Wl,···,W J. l.

< P P

is contained in RFG •

By taking Q :::

arr

E R where ITl is the projection 1 4>1""

I4>P/Wl/" ,W

_p

operator onto the orthocomplement of span <~1/

••• ,4>p/W1""'Wp>

and

8

-Remarks. The strong topology on B(H) restricted to the amplification of

B(X) is equal to the ultra-strong topology on B(X).

By

Dixmier [D], the ultra-strong topology on B(X) is strictly finer than the strong topology

on

B(X)

iff

X

is infinite dimensional. Notice also that the ultra-strong

and the strong topology coincide on bounded sets.

THEOREM 5

The reduction-amplification mapping

*

P

B(H) -+ B(H) c

(i) is not continuous with respect to the strong topology on B(H}. (ii) is not continuous with respect to the weak topology on B(H) . PROOF:

(i) Consider the set LF

=

{BIB

=

Q ® I, Q

E

SF} C B(H) with SF as in Lemma 4. Suppose there exists a finite sequence (Ql""'~p) cHand

*

(ii)

o

> 0 such that

P

maps the set c

into L F' We look at operators of the form A = Q ® PO' Q

E

B(X) . 00

Write ~i

₌

~ ~ik

® _{vk · Then} II (Q ® _{POH\II H '"} IIQ4>iO ® voliH =

k=O

=

IIQ4>iOIlX- So i f the operators Q are in the set

_S~10~""~PO'

the operators A

=

Q ® Po are in U

<P

1,· .. ,<lip•

For these special operators we have AO

=

Q. SO it follows from Lemma 4

*

that

P

does not map

U

m ~ , as a whole, into L_F•

c w1" •. ,w_p

Consider the set -FG

=

{BIB

=

Q ® I, Q

E R

_FG} c B(H) with

R

_FG as in Lemma 4. Suppose there exist finite sequences (<P₁"",<li_{p )'} (~l""'~p) c

*

cHand IS > 0 such that

P

maps the set c

9

-into E_FG. Again we look at operators A of the form A = Q ® PO' Q € B (X) • Write '¥i

=

So if the operators Q are in the set

R

,lo I

$10""'~pO,1jJ10""'~10 then the operator A = Q 0 Po belongs to V

4>l,···,4>p,\jIl'···''¥p· For these special operators we have A = Q ® Po'

;.

So it follows from Lemma 4 that

P

does not map the set c

V as a whole, into ~FG'

<i> l' .' .. , ~ p' 'Y 1 ' ••. , \jI p'

REFERENCES

[D], Dixmier, J. Von Neumann Algebras North Holland, Amsterdam, etc. 1981. "[M] Muynck, W.M. de, A Quantum Mechenical Theory of Local Observables

and Local 'Operators. Foundations of Phy~ics!! (1984).

[N] Naimark, M.A. Normed Algebras Wolters-Noordhoff Publ. Groningen 1970. [W] Weidmann, J. 'Linear Operators in Hilbert spaces. G.T.M. Vol. 68,

Springer-Verlag, Berlin, etc. 1980