Joint Spatial Numerical Range of Tensor ProductsAuthor(s): Ram VermaSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 90A, No. 2 (1990), pp. 201-204Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20489358 .

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JOINT SPATIAL NUMERICAL RANGE OF TENSOR PRODUCTS

By RAM VERMA

Department of Mathematics, Duke University*

(Communicated by T. T. West, M.R.1.A.)

[Received 5 July 1989. Read 1 October 1990. Published 31 December 1990.]

ABSTRACT

We prove that the joint spatial numerical range of the tensor product of the bounded Banach space operators equals the Cartesian product of their usual spatial numerical ranges.

1. Introduction

Let X, Y be two complex Banach spaces, and let X 0 Y denote the algebraic

tensor product. For an arbitrary complex Banach space Z, let L(Z) denote the

algebra of all bounded linear operators on Z.

1.1 Definition. A norm a or 11,,, on X0 Y is called a cross-norm if

a (x Oy) = IIx X Yii =

Ixjj IjyjI

for every x 0 y E X 0 Y.

1.2 Definition. A norm a on X0 Y is called a reasonable cross-norm if a is a

cross-norm on X0 Y and the dual norm a' induced by the dual (XX Y)* is also a cross-norm on X * X Y *.

1.3 Definition. A norm a on X0 Y is called a tiniform cross-norm if a is a cross

norm on X0 Y, and if

a ((S 0 T)z) ? SIJ || T|la (z)

for every SeL(X), TEL(Y), zEX(Y.

From now onward, we consider only the case of a uniform reasonable cross

norm. Let X 0Y denote the completion of X0(D Y with respect to norm a. Then, if there are more than two Banach spaces involved, the uniform reasonable cross

norms are assumed to be associative. Thus, X = Xi OX2 0X OX, means the

*Present address: University of Central Florida, Orlando, Florida 32816.

Proc. R. Ir. Acad. Vol. 90A, No. 2, 201-204 (1990)

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202 Proceedings of the Royal Irish Academy

completion of X1 0 X2 0 ..08 X, with respect to an associative uniform reason

able cross-norm a. For details about cross-norms see [4], [6], [3] and [8]. Next, we consider the operators AjE L(Xj), and define the operators A?

(1 ? j G n) acting on X by

A? = I, 01 j1X, A0 Ij+1 , (1)

where Ij is the identity operator on X1.

1.4 Definition. For an arbitrary complex Banach X, and A= (AI, A2,. . , A,) E L(Xf) we define the joint spatial numerical range V(A) of

A by

V (A) = {(f (A Ix), f (A2x), . . . f(A,,x)): 1 = lIfIl = lixl =

f (x)}, (2)

wherefEX* and xEX. For details about the joint spatial numerical range, see [1] and [8].

We observe that

V(Al, A2,... 7An) c V(A1) x V(A2) x ... x V (A,,). (3)

Dash [2] has shown that the joint numerical range of the tensor product of the

bounded Hilbert space operators is the same as the Cartesian product of their

usual numerical ranges. The aim of this paper is to prove an analog of this result

to the case of the bounded Banach space operators on the tensor products.

Although the initial motivation for this problem was developed as a result of the

recent publications of Rynne [5] and Wrobel [8], we expect that it would be a

nice result on the joint spatial numerical ranges of the tensor products.

2. Joint spatial numerical ranges

We are just about ready to prove the main result on the joint spatial numerical

ranges.

2.1 Theorem. Let A? - (A?, AO,.. . ,AO) be an n-tuple of operators acting on

i= X1b x2* 0 ' 0)X,, as defined by (1). Then, for 1 l j ?n, we have

V(AO,Ar A@) = V(A1) x V(A2) x ; x V(A,. (4)

PROOF. In order to prove the inclusion

V (A A , A@) C V (A 1) X V (A2) X ...X V (A,,),

we first prove that V(A() = V(Aj) (j = 1, 2, ... , n). Since this follows by induc

tion from the case n - 2, we prove that

V(A10 I2) = V(13 0 Al) = V(A1).

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VERMA -Joint spatial numerical range of tensor products 203

To prove V(A1 0 I2) = V(A1), let us assume that A L V(A1). Then, for every

x1 E X1, there exists a linear functional f e Xt such that A =f (Aix). Suppose, for some x2 E X2, that there exists a linear functional g L X 2 such that 1 = 11x211 =

g11 = g(x2). If we set u = xI O X2, then it is clear that u & X10 X2. Since

f 0 g((A 01 I2)u) =f 0 g(A,x, t X2)-f (A lxl) = A,

this implies that V(A1) C V(A1 0I2). Conversely, assume A E V(At I2). Then, for some v L l I0X2, there exists

a linear functional h on X1 OX2 such that

h((AI 0 12)v) = A.

Now, for some A E L(XI) and x E Xi, we define a linear functional f on XI such that f(Ax) = h((A 0 I2)v). Then we find that

If(Ax) = h((A 0 I2)v)l l lhil ||A ||llvii,

f (Ilx) = h((I 0 I2)v)=1,

lflf sup If(Ax)i= sup Jh((A 0I2)v) IJA?KS ' IA?I211h I

? lIhIl Ilvil = 1, and lIf I = 1.

From this, it follows that

A = h((A1 I12)v) =f(A lx) E V(A,).

The proof of V(13 0 A1) = V(A1) is similar to that of the first part. Hence, in

view of the inclusion (3), this establishes the inclusion of the theorem one way. To prove the converse inclusion, let us assume that

A = (At A2 . A,) E V(A3) x V(A2) X x V(A,).

Then, for some xj E Xj, there exists an fj E X with 1 llxj1l = 11fj1l = fj(xj) such

that Aj = fj(Ajxj), for all j, 1 j -

n.

Set u? = xi0 .0xI0xxj,0x I0 0 x,L X^, and

f?=fl0 0fj 3-10fj0f10 .. 02)tk E X.

Then 1 = Il|k= f?Ila =f?(u?), and Aj =f(A0u?) =fj(Ajxj), for all j, 1 ?Zj s n. From this, it follows that A E V(AV, A"I.' A). 1Hence,

V(A1) x V(A2) X .. x V(Ac) C V(AO, A . , A?) U

2.2. Remark. Let JT(A?) denote the Taylor spectrum [7] of AO =

(AO, AO. A$). Then, in the light of the work of Rynne [5, theorem 1], we

establish the inclusion relation

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204 Proceedings of the Royal Irish Academy

cTT(A, A, . . . ,A C)

V (At, AO, . . . AO)-,

where the bar denotes the closure.

ACKNOWLEDGEMENT

The author would like to thank the referee for the valuable suggestions leading

to the revised version.

REFERENCES

[1] Ch?, M. 1986 Joint spectra of operators on Banach space. Glasgow Math. J. 28, 69-72.

[2] Dash, A. T. 1973 Tensor products and joint numerical range. Proc. Am. Math. Soc. 40, 521

6.

[3] Eschmeier, J. 1988 Tensor products and elementary operators. J. Reine Angew. Math. 390,

47-66.

[4] Ichinose, T. 1970 On spectra of tensor products of linear operators in Banach spaces. J. Reine

Angew, Math. 244, 119-53.

[5] Rynne, B. 1987 Tensor products and Taylor's joint spectrum in Banach spaces. Proc. R. Ir.

Acad. 87A, 83-9.

[6] Schatten, R. 1950 A theory of cross-spaces. Princeton University Press.

[7] Taylor, J. 1970 A joint spectrum for several commuting operators. J. Fund. Anal. 6, 172-91.

[8] Wrobel, V. 1986 Tensor products of linear operators in Banach spaces and Taylor's joint

spectrum. J. Operator Theory 16, 273-83.

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