Projections on tensor products of Banach spaces

c 2008 Birkh¨auser Verlag Basel/Switzerland 0003/889X/040341-12,published online2008-03-20

DOI 10.1007/s00013-007-2350-9 Archiv der Mathematik

Projections on tensor products of Banach spaces

Fernanda Botelho and James Jamison

Abstract. We characterize norm hermitian operators on classes of tensor prod- ucts of Banach spaces and derive results for the particular settings of injective and projective tensor products. We provide a characterization of bi-circular and generalized bi-circular projections on tensor products of Banach spaces supporting only dyadic surjective isometries.

Mathematics Subject Classification (2000).Primary 30D55; Secondary 30D05.

Keywords. Isometry, bi-circular projections, generalized bi-circular projec- tions, injective and projective tensor products of Banach spaces.

1. Introduction. In this paper, we characterize the structure of norm hermitian operators on tensor products of Banach spaces in which the only surjective isome- tries are of dyadic type. Khalil in [9], Khalil-Salem in [8], and Jarosz in [11] pro- vided classifications of surjective isometries for different tensor products of Banach spaces that assure the existence of spaces with such isometries. The structure of norm hermitian operators allows an easy characterization of those operators that are also hermitian projections. Such characterization can be transcribed for bi- circular projections, as established by Jamison in [10]. The last section extends previous results to the more general case of generalized bi-circular projections, introduced in [7], and provides characterizations of these projections in a variety of tensor product spaces. Characterizations of generalized bi-circular projections in various Banach spaces can be found in [3], [4] and [13].

We start by recalling the definitions of norm hermitian operators, bi-circular and generalized bi-circular projections, see [6] and [7].

Definition 1.1. We consider a complex Banach spaceX. A bounded operatorSon X is said to be hermitian if and only if{e^itS}t∈R defines a one-parameter group of isometries. An operator P on X is said to be a bi-circular projection if and only ifP²=P andP+λ(Id−P) is an isometry for every complex numberλof

modulus 1. An operatorP_λ(onX) is said to be a generalized bi-circular projection if and only ifP_λ²=P_λ andP_λ+λ(I−P_λ) is an isometry ofX, for someλ∈C, λ= 1, and|λ|= 1.

We observe that such isometries must be surjective. In fact, if ω ∈ X, there existsz∈X, z=P_λ(ω) +_λ¹(ω−P_λ(ω)),such that [P_λ+λ(I−P_λ)](z) =ω.

We consider the algebraic tensor product of two Banach spaces X₁ and X₂, denoted by X₁⊗X₂, equipped with some crossnorm α, cf. [12]. We denote the completion ofX₁⊗X₂relatively to this crossnorm byX₁⊗αX₂.The two most well- known crossnorms onX₁⊗X₂are the so called projective crossnorm (denoted byν) and injective crossnorm (denoted byλ). The corresponding completions relative to these norms are called projective and injective tensor products, commonly denoted byX₁⊗ˆX₂andX₁⊗ˇX₂,respectively. For completeness of exposition, we recall that the projective tensor norm is defined as follows

ν(z) = inf _n

i=1

x_i y_i: z= n i=1

x_i⊗y_i

the injective tensor norm is defined as follows λ

_n

i=1

x_i⊗y_i

= sup

| n i=1

ϕ(x_i)ψ(y_i)|,ϕ=ψ= 1

.

It is shown in [12, 5] thatλis the least crossnorm andµthe greatest crossnorm, i.e. for every reasonable crossnormαonX₁⊗X₂,andz∈X₁⊗X₂, we have that

λ(z)≤α(z)≤µ(z).

Definition 1.2. We say that a bounded operator T onX₁⊗αX₂ is dyadic if and only if there exist bounded operators on the component spaces, denoted byT₁and T₂,so thatT =T₁⊗T₂.

It follows from the Hahn Banach theorem that the representation of a dyadic operator as the tensor product of two factors is essentially unique. If T₁⊗T₂ = T₁⊗T₂ then there must exist a scalaraso thatT₁=aT₁ andT₂=aT₂.Moreover, given a dyadic isometryT₁⊗T₂,T₁ is an isometry if and only ifT₂ is an isometry.

2. Norm Hermitian Operators on Tensor Products Spaces with Dyadic Isometries.

In this section we characterize the norm hermitian operators on a tensor product of two Banach SpacesX₁⊗αX₂, whereαis a reasonable crossnorm.

Theorem 2.1. If every surjective isometry onX₁⊗αX₂is dyadic, thenSis a norm hermitian operator onX₁⊗αX₂ if and only if either

1. S=rId_X1⊗αX2,for somer∈R,or

2. There exist hermitian operators L and R,on X₁ and X₂ respectively, such thatS(x₁⊗x₂) =L(x₁)⊗x₂+x₁⊗R(x₂).

Proof. IfSis either of form (1) or (2) then we show that it is an hermitian operator.

It is sufficient to prove that T_t = e^itS is a one-parameter group of isometries.

This follows trivially, whenever S is a multiple of the Id, since T_t = e^ritId. If S(x₁⊗x₂) = L(x₁)⊗x₂+x₁⊗R(x₂) then T_t(x₁⊗x₂) = e^itL(x₁)⊗e^itR(x₂). Each tensor factor is a one-parameter group of isometries, so is{T_t}.

Conversely, given S, an hermitian operator on X₁⊗αX₂, then T_t = e^itS is a uniformly continuous one-parameter group of isometries onX₁⊗αX₂, see [6]. Each isometry is dyadic, hence T_t = L_t⊗R_t, with L_t and R_t surjective isometries on X₁ and X₂ respectively. We show that these components also define uniformly continuous one-parameter groups of isometries. We assume that L₀ =Id_X1 and R₀=Id_X2. Furthermore, we first assume that{T_t}is a nontrivial family, i.e. for everyt= 0, T_tis not a multiple of theId.

Step I{L_t}and{R_t}are uniformly continuous families of operators.

T_t−T_t0= sup{α((T_t−T_t0)(z)) : α(z) = 1}

≥ sup{α[(T_t−T_t0)(x⊗y)], x=y= 1}

= sup{α[(L_t(x)⊗R_t(y)−L_t0(x)⊗R_t0(y)], x=y= 1}

≥ sup{λ[L_t(x)⊗R_t(y)−L_t0(x)⊗R_t0(y)], x=y= 1}

= sup{ϕ[L_t(x)−L_t0(x)]R_t(y) +ϕ(L_t0(x))[R_t(y)−R_t0(y)]X2, x=y=ϕ= 1}.

We assume that there exists x ∈ X₁ (depending ont) of norm equal to 1 such that{L_t(x)−L_t0(x), L_t0(x)}is linearly independent. The Hahn-Banach theorem asserts the existence ofϕ∈X^∗ such thatϕ(L_t0(x)) = 1, ϕ(L_t(x)−L_t0(x)) = 0, andϕ= 1.Therefore

T_t−T_t0 ≥sup{R_t(y)−R_t0(y)X2 : y= 1}=R_t−R_t0.

Now, we assume the existence of a sequence {t_n} converging to t₀ such that for everynandx∈X₁,{L_tn(x)−L_t0(x), L_t0(x)} is linearly dependent. This means that L_tn(x)−L_t0(x) = a_n(x)L_t0(x), for scalarsa_n(x) depending on both t_n and x. We prove that each function a_n(x) is, in fact, independent of x. We start by selecting two linearly independent vectors inX₁, say x₁ andx₂ (X₁ andX₂ are of dimension greater than 1). Therefore

L_tn(x₁+x₂)−L_t0(x₁+x₂) =a_n(x₁+x₂)L_t0(x₁+x₂)

=a_n(x₁)L_t0(x₁) +a_n(x₂)L_t0(x₂),

anda_n(x₁) =a_n(x₁+x₂) =a_n(x₂).On the other hand, forx₁=kx₂ (k a scalar) we have thata_n(x₁) = a_n(x₂), henceL_tn = (a_n+ 1)L_t0, with |a_n+ 1| = 1. For

eachn,we have

T_tn−T_t0 ≥ sup{α(L_tn(x)⊗R_tn(y)−L_t0(x)⊗R_t0(y)) : x=y= 1}

= sup{α((a_n+ 1)L_t0(x)⊗R_tn(y)−L_t0(x)⊗R_t0(y)) : |x=y= 1}

= sup{L_t0(x)X1(a_n+ 1)R_tn(y)−R_t0(y)X2, x=y= 1}

= sup{(a_n+ 1)R_tn(y)−R_t0(y)X2 : y= 1}

= sup{|(a_n+ 1)ψ(R_tn(y)−ψ(R_t0(y))|: y=|ψ= 1}.

Moreover, if there exists a y_n ∈X₂ (of norm 1) such that {R_tn(y_n), R_t0(y_n)} is linearly independent, then let ψ ∈ X₂^∗ such that ψ(R_tn(y_n)) = a_n+ 1 and ψ(R_t0(y_n)) = 0. This would imply that sup{|(a_n + 1)ψ(R_tn(y))−ψ(R_t0(y))| : y = |ψ = 1} = 1 and then T_tn −T_t0 ≥ 1. This leads to a contradiction, since{T_t} is uniformly continuous. Therefore we assume that for every n andy, {R_tn(y), R_t0(y)} is linearly dependent. As previously shown, there exist scalars depending ont_n so thatR_tn= (b_n+ 1)R_t0 (|b_n+ 1|= 1).Since we also have that L_tn = (a_n+ 1)L_t0,thenT_tn−tm= _(a^(an+1)(bn+1)

m+1)(bm+1)Id. Consequently, there must exist a sequence {τ_n}, converging to zero, and modulus 1 complex numbersλ_n, such thatT_τn =λ_nId, equivalentlye^{i τnS} =e^ln(λn)Id. Since the operatorSis hermitian, it has real spectrum (σ(S)), the spectrum of ln(λ_n)Id is clearly ln(λ_n).Theorem 6, in [16], implies thatλ_n= 1 orS−ln(λ_n)Id = (2k_nπi)Id, for some integersk_n. In either caseT_τn is a multiple of the identity, contradicting our initial assumption.

We have shown that{R_t}is a uniformly continuous family of surjective isome- tries. Now we prove that{L_t}is also uniformly continuous. For every >0 there exists δ > 0 so that given t with |t−t₀| < δ, we have T_t−T_t0 < /2 and R_t−R_t0< /2.Consequently, we have that

sup{R_t(y)−R_t0(y)X2 : y= 1}=R_t−R_t0< /2 and

T_t−T_t0

= sup_x=y=|ϕ=1{ϕ[L_t(x)−L_t0(x)]R_t(y) +ϕ(L_t0(x))[R_t(y)−R_t0(y)]_X2}

≥sup_x=y=|ϕ=1{|ϕ[L_t(x)−L_t0(x)]|R_t(y) − ϕ(L_t0(x))/2}

=L_t−L_t0 −/2.

ThereforeL_t−L_t0< and the uniform continuity of{L_t}follows as well.

Step II{L_t} and{R_t}are weakly differentiable.

We observe that the functionf(t) =T_tis strongly differentiable, hence weakly differentiable. For every functional Φ∈(X₁⊗αX₂)^∗ we have that

t→tlim0

Φ

T_t(z)−T_t0(z) t−t₀

exists.

In particular, this limit also exists for functionals of the formϕ⊗ψ. The linearity of f allows us to reduce the problem to the differentiability at zero. Hence, for z=x⊗y,we have

h→0limϕ⊗ψ

Th(z)−z

h = lim

h→0ϕ⊗ψ

(Lh⊗Rh)(x⊗y)−x⊗y h

= lim

h→0ϕ⊗ψ

L_h(x)⊗^Rh(y)−y_h +^Lh(x)−x

h ⊗y

.

If there existsy,so that{^Rh(y)−y_h , y}is linearly independent, letψbe a functional onX₂ that attains the value 1 aty and 0 at ^Rh(y)−y_h .In this case, we have that

h→0lim ϕ⊗ψ

L_h(x)⊗R_h(y)−y

h +L_h(x)−x h ⊗y

= lim

h→0ϕ

L_h(x)−x h

. Thereforeg(t) =L_tis weakly differentiable. Similarly, if we assume the existence of x such that {^Lh(x)−x_h , x} is linearly independent it follows that h(t) = R_t is weakly differential. The weak differentiability of either g(t), or h(t) implies the weak differentiability ofh(t), org(t) respectively. It remains to consider the exis- tence of a sequenceh_n converging to zero such that for everyx∈X₁ andy∈X₂, {^Lh(x)−x_h , x}and{^Rh(y)−y_h , y}are both linearly dependent. An analogue of a pre- vious argument would imply thatT_t is trivial, for some values oft, contradicting our assumption.

Step III{R_t}and{L_t} are one parameter groups of isometries.

The group conditionT_t1+t2 =T_t1◦T_t2 implies thatL_t1+t2=λ(t₁, t₂)L_t1◦L_t2 and R_t1+t2 = λ(t₁, t₂)R_t1 ◦R_t2, for some modulus 1 scalars. We prove that λ(t₁, t₂) = 1, for every t₁ and t₂. We recall that T₀ = Id_X1⊗αX2 = Id_X1⊗Id_X2 and without loss of generality we may assume that L₀ =Id_X1 and R₀ =Id_X2. We also have that L₀ = Id_X1 = λ(t₁,−t₁)L_t1 ◦ L_−t1 = λ(−t₁, t₁)L_−t1 ◦ L_t1 and L_t1 = λ(t₁,−t₁)L⁻¹_−t

1 implying that Id_X1 = λ(−t₁, t₁)L_−t1 ◦ L_t1 = λ(−t₁, t₁)λ(t₁,−t₁)L_−t1◦L⁻¹_−t

1. Thereforeλ(−t₁, t₁) =λ(t₁,−t₁) andL_−t1◦L_t1 = L_t1◦L_−t1.

We clearly haveλ(0, t) =λ(t,0) = 1,for allt.

First, we observe thatλ(t₁, t₂) = λ(t₂, t₁) if and only ifL_t1◦L_t2 =L_t2◦L_t1. In order to prove this last statement we proceed as follows:

L_3t=λ(2t, t)L_2t◦L_t=λ(2t, t)λ(t, t)L_t◦L_t◦L_t=λ(2t, t)L_t◦L_2t and

L_t◦L_2t=L_2t◦L_t.

This last statement is equivalent to λ(2t, t) = λ(t,2t). Inductively we show that L_mt◦L_nt =L_nt◦L_mt and λ(nt, mt) =λ(mt, nt), forn, m integers and t a real number. Therefore we haveL_r1◦L_r2 =L_r2◦L_r1 for rational valuesr₁andr₂and continuity implies thatL_t1◦L_t2 =L_t2◦L_t1 andλ(t₁, t₂) =λ(t₂, t₁).

Furthermore, for arbitrary values of t, say t, t₁, t₂ we have that λ(t + t₁, t₂)λ(t, t₁) =λ(t₁+t₂, t)λ(t₁, t₂).The weak differentiability established in Step II implies the differentiability ofλ,then we have

∂_tλ(t+t₁, t₂)λ(t, t₁) +λ(t+t₁, t₂)∂_tλ(t, t₁) =∂_tλ(t, t₁+t₂)λ(t₁, t₂).

Hence, fort=t₂, the equation above implies that∂_tλ(t₂, t₁) = 0 andλ(t₂, t₁) = C(t₁), a constant depending ont₁. For t₂ = 0, we have that 1 =λ(0, t₁) =C(t₁) and we have established thatλ= 1 which completes the proof of Step III.

The families {L_t} and {R_t} are one-parameter groups of uniformly continu- ous family of isometries, hence there exist hermitian operators L and R so that L_t=e^itL andR_t=e^itR.Therefore we have that T_t =e^itS =e^itL⊗e^itR and the corresponding generator satisfies

S=

−i d dt

t=0

e^itS(x₁⊗x₂) =L(x₁)⊗x₂+x₁⊗R(x₂).

This completes the proof of statement 2, provided that{T_t}is a nontrivial family.

If we assume that, for somet₀, T_t0 is a multiple of the Id, then Theorem 6, in [16], implies thatλ= 1 or S−ln(λ)Id = (2kπi)Id, for some integerk. In either caseS is a multiple of the identity. This completes the proof of the theorem.

3. Norm Hermitian Operators on Projective and Injective tensor Products. The- orems by Khalil and Saleh [8, 9] state that surjective isometries on a class of projective tensor products are dyadic. A theorem by Jarosz states that surjective isometries, that are not reflections, on a class of injective tensor products are also dyadic. This isometry structure and the theorem 2.1 provide the infrastructure for the characterization of norm hermitian operators on Khalil-Saleh projective tensor products and Jarosz injective tensor products, as it will be shown in the forthcoming corollary 3.3. We start by stating Khalil, Khalil-Saleh and Jarosz characterizations.

Theorem 3.1. 1. (Khalil in [9]) T is a surjective isometry on L^p⊗ˆL^p (p > 1) if and only if there exists surjective isometries T₁, T₂ on L^p such that T = T₁⊗T₂.

2. (Khalil and Saleh in[8]) If X andY are an ideal pair of Banach spaces i.e.

X and Y are reflexive Banach spaces so thatX and Y^∗ are strictly convex andX^∗ has the approximation property ([5]), then every surjective isometry T on X⊗ˆY^∗ is of the form T =T₁⊗T₂, for surjective isometriesT₁, T₂ on X andY^∗.

Theorem 3.2. (Jarosz in[11]) IfX₁is a complex Banach space with trivial central- izer andX₂a complex Banach space with strictly convex dual, then every isometry T fromX₁⊗ˇX₂ onto itself is of the form

1. T(x₁⊗x₂) =T₁(x₁)⊗T₂(x₂), where T₁ andT₂ are onto isometries.

2. There exists a Banach space Z such that Z⊗ˇX₂ is isometric to X₁ and T under this identification is of the formT(z⊗a⊗b) =z⊗b⊗a, for allz∈Z anda, b∈X₂.

Corollary 3.3. LetE=E₁⊗_αE₂ with E_i of any of the following forms:

1. E₁=E₂=L^p andα=ν,

2. E₂=Y^∗ where(E₁, Y)is an ideal pair of Banach spaces andα=ν, or 3. E₁a Banach space with trivial centralizer andE₂a Banach space with strictly

convex dual andα=λ,

thenS is a hermitian operator on E if and only if either 1. S=rId , for somer∈R, or

2. There exist hermitian operators L on E₁, and R, on E₂, respectively, such thatS(x₁⊗x₂) =L(x₁)⊗x₂+x₁⊗R(x₂).

Proof. IfSis of any of the forms (1) or (2) then it is clearly hermitian as previously shown. IfSis hermitian then{e^itS}tis a one-parameter group of isometries. This situation follows clearly from the Theorem 2.1, provided that for every t, e^itS is a dyadic isometry. On the other hand, if there exists a t₀(= 0) so that e^it0S is not dyadic then there must exist an isometry ontoZ⊗ˇX₂ such that,e^it0S under this identification is of the form described in Jarosz theorem. This implies that e^2it0S =Id therefore S is a multiple of the identity, see [16]. This completes the

proof.

4. Bi-circular Projections on Injective and Projective Tensor Products. The notion of bi-circular projection on a Banach space was first introduced by Sta- cho and Zalar in [17] and [18]. A projectionP on a Banach spaceX is said to be bi-circular ife^iaP+e^ib(I−P) is an isometry for all choices of real numbersaand b. These projections are in fact norm hermitian, as shown in [10]. The following theorem characterizes these projections in a class of tensor product spaces.

Theorem 4.1. If every surjective isometry on X₁⊗αX₂ is dyadic, thenS is a her- mitian projection onX₁⊗ˆX₂ if and only ifS=Id_X1⊗R orL⊗Id_X2 whereL and R are hermitian projections onX₁ andX₂, respectively.

Proof. We begin by assuming thatSis a hermitian projection then by the previous theorem

S(x₁⊗x₂) =L(x₁)⊗x₂+x₁⊗R(x₂),

where L and R are hermitian operators on X and Y respectively. Since S is a projection then

[L²−L](x₁)⊗x₂+ 2L(x₁)⊗R(x₂) +x₁⊗[R²−R](x₂) = 0.

(4.1)

Hahn-Banach Theorem leads to a contradiction if there existsx_i ∈X_i (i= 1,2) so that either {x₁, L(x₁), [L² − L](x₁)} or {x₂, R(x₂),[R² − R](x₂)} is lin- early independent. Therefore for every x_i ∈ X_i, {x₁, L(x₁),[L²−L](x₁)} and {x₂, R(x₂),[R²−R](x₂)} are linearly dependent. If there existsx₁ = 0 so that L(x₁) =ax₁ then equation 4.1 reduces to

x₁⊗

(a²−a)x₂+ 2aR(x₂) + (R²−R)(x₂)

= 0,

andR²+ (2a−1)R+ (a²−a)Id≡0.A theorem due to Taylor (see [15]) applied to Rimplies thatR=−a P₁+ (1−a)P₂=P₂−aId,whereP₁andP₂are projections such thatP₁◦P₂=P₁◦P₂= 0 andais a real number. These projections are also hermitian projections. The equation 4.1 reduces to

(L²−L)(x₁)−2aL(x₁) + (a²+a)x₁

⊗x₂+ [2L(x₁)−2ax₁]⊗P₂(x₂)≡0.

Therefore, for x₂ in the range of P₂, the last equation implies that L²+ (1− 2a)L+ (a²−a)Id= 0 andL=aQ₁+ (a−1)Q₂, withQ₁andQ₂two orthogonal projections. SinceS(x₁⊗x₂) =−Q₂(x₁)⊗x₂+x₁⊗P₂(x₂),equation 4.1 now implies thatQ₂(x₁)⊗P₂(x₂) = 0 and then eitherQ₂ orP₂ is the zero projection. Clearly ifS =Id_X1⊗R or L⊗Id_X2, with L and R hermitian projections onX₁ and X₂ respectively,S is an hermitian projection. This completes the proof.

Remark 4.2. 1.S is a hermitian projection on L^p⊗ˆL^p (p >1) or on a projective tensor productX⊗ˆY^∗where (X, Y) is an ideal pair of Banach spaces, if and only if S =Id_X⊗S_Y or S_X⊗Id_Y where S_X and S_Y are hermitian projections on X andY respectively.

2.S is a hermitian projection onX₁⊗ˇX₂, withX₁a Banach space with trivial centralizer andX₂ a Banach space with strictly convex dual, if and only if S = Id_X1⊗S_X2 orS_X1⊗Id_X2 whereS_X2 andS_X1 are hermitian projections onX₂and X₁ respectively.

5. Generalized Bi-Circular Projections as Dyadic Operators. A generalization of bi-circular projections was recently introduced by Fosner, Illisevic, and Li in [7].

It concerns with projectionsP_λ onX so thatP_λ+λ(I−P_λ) is an isometry ofX, for some modulus 1 complex numberλ= 1.

In the next theorem B(X₁⊗αX₂) represents the bounded operators on X₁⊗αX₂.

Theorem 5.1. If λ∈T with λ= 1, then P_λ ∈ B(X₁⊗αX₂) is a projection asso- ciated with a dyadic isometry (i.e.P_λ+λ(I−P_λ)is dyadic) if and only ifP_λ is dyadic.

Proof. Given a generalized dyadic projectionP_λ,the isometryP_λ+λ(Id−P_λ) is clearly dyadic.

We prove the converse, if T denotes a dyadic isometry associated with P_λ, thenP_λ is also dyadic. SinceP_λ is a projection thenT must satisfy the algebraic equationT²−(λ+ 1)T+λI= 0.FurthermoreT =T₁⊗T₂,hence we have that

T₁²(x)⊗T₂²(y)−(λ+ 1)T₁(x)⊗T₂(y) +λx⊗y= 0, for allx∈X₁,andy∈X₂, (5.1)

withx⊗y interpreted as an operator from the dualX₁^∗intoX₂.For everyϕ∈X₁^∗ the equation (5.1) yields

ϕ(T₁²(x))T₂²(y)−(λ+ 1)ϕ(T₁(x))T₂(y) +λϕ(x)y= 0.

(5.2)

We first assume thatλ=−1,then (5.2) reduces toϕ(T₁²(x))T₂²(y) =ϕ(x)y.For each x ∈ X₁ we have that T₁²(x) = a_xx (for some scalar a_x) and T₂² = a_xId. The linearity of T₁ implies that a_x is independent of x. Hence T₁² = aId and T₂² =aId. ThereforeP_λ = ^Id+T₂ =Id, which is clearly dyadic. Now, we assume λ=−1.If, in addition, there existsx₁∈X₁so that{x₁, T₁(x₁), T₁²(x₁)}is linearly independent then the Hahn-Banach theorem assures the existence of a functional in X₁^∗ such that ϕ(T₁(x₁)) = 1 and ϕ(T₁²(x₁)) = ϕ(x₁) = 0. This leads to a contradiction. Hence, for allx ∈ X₁, the set {x, T₁(x), T₁²(x)} must be linearly dependent. If there existsx∈X₁such that{x, T₁(x)}is linearly independent then T₁²(x) =a x+bT₁(x),for some scalarsaandb. Then equation (5.2) reduces to

[aϕ(x) +bϕ(T₁(x))]T₂²(y)−(λ+ 1)ϕ(T₁(x))T₂(y) +λϕ(x)y= 0.

We select a functional ϕsuch thatϕ(x) = 1 andϕ(T₁(x)) = 0. Hence aT₂²(y) + λy = 0, for all y ∈ X₂. This implies that T₂² = cId, for some c of modulus 1. We also select a functional ψ such that ψ(x) = 0 and ψ(T₁(x)) = 1. Then b c y−(λ+ 1)T₂(y) = 0 and T₂ = dId for some scalar d of modulus 1. The equation (5.2) becomes φ(d²T₁²(x)−(λ+ 1)d T₁(x) +λx) = 0, for all φ ∈ X₁^∗. Therefored²T₁²−(λ+ 1)d T₁+λId = 0. The projectionP_λ is given as follows

P_λ(x⊗y) = 1

1−λ [−λx⊗y+T₁(x)⊗T₂(y)] = 1

1−λ [−λx+d T₁(x)]⊗y.

We set S₁(x) = ^{−λx+d T}_1−λ^1(x), hence P_λ = S₁ ⊗ Id. The remaining case assumes that for everyx∈X₁,{x, T₁(x)}is linearly dependent. For eachx,there exists a modulus 1 scalare_x such thatT₁(x) =e_xx. The linearity ofT₁assures that e_xis independent ofxand thenT₁=eId. The equation (5.2) becomes

e²φ(x)T₂²(y)−(λ+ 1)eφ(x)T₂(y) +λφ(x)y= 0,

for every φ ∈ X₁^∗ and y ∈ X₂. Therefore e²T₂²−e(λ+ 1)T₂+λId = 0 and

P_λ=Id⊗S₂ withS₂(y) =^−λy+eT_1−λ^2(y).

It is a consequence of the previous proof the following corollary.

Z₁⊗αZ₂ S -

Z₁⊗αZ₂ U₁⊗U₂

? ?U₁⊗U₂ X₁⊗αX₂ T - X₁⊗αX₂

Figure 1. S andT are tensor conjugate if and only if the diagram commutes.

Corollary 5.2. Ifλ=−1, P_λ∈ B(X₁⊗αX₂)is a generalized bi-circular projection associated with a dyadic isometry if and only ifP_λ is either of the formS₁⊗Id or Id⊗S₂, withS_i a generalized bi-circular projection onX_i.

Proof. It was shown in the previous that ifP_λis associated with a dyadic isometry then

P_λ(x⊗y) =S₁(x)⊗y or P_λ(x⊗y) =x⊗S₂(y)

withS₁(x) = ^{−λx+d T}_1−λ^1(x) and S₂(y) = ^{−λy+e T}_1−λ^2(y).The operators S₁ and S₂ are generalized bi-circular projections on the component spaces since d²T₁²−(λ+ 1)d T₁+λId = 0 and e²T₂²−e(λ+ 1)T₂+λId = 0. It is trivial to check the

converse.

Definition 5.3. Given the Banach spacesX₁,X₂,Z₁andZ₂, we consider the tensor productsX₁⊗αX₂ and Z₁⊗αZ₂,representing the completions of X₁⊗X₂ and Z₁⊗Z₂relative to the crossnormα. A bounded operatorSonZ₁⊗αZ₂is said to be tensor conjugateto a bounded operator T, onX₁⊗αX₂, if and only if there exists a dyadic isometryU₁⊗U₂ with isometric factorsU_i : Z_i → X_i such that (see Figure 1)

T = (U₁⊗U₂)◦S◦(U₁⁻¹⊗U₂⁻¹).

(5.3)

Remark 5.4. Isometric properties are preserved under tensor conjugacy. The equa- tion 5.3 also implies thatT^k = (U₁⊗U₂)◦S^k◦(U₁⁻¹⊗U₂⁻¹),for every positive inte- gerk. In particular, we conclude that, forX₁=X₂,the operatorS(a⊗b) =b⊗a is not tensor conjugate to a dyadic one.

If X₁ has trivial centralizer (see [1] for the definition) and X₂ has strictly convex dual it was shown in [11] that there exists a Banach space Z so that the injective tensor product Z⊗ˇX₂ is isometric to X₁, we denote such isometry by U. IfT is a nondyadic surjective isometry on X₁⊗ˇX₂ then S =U⊗Id_X2 ◦ T ◦ U⁻¹⊗Id_X2,acting on the basis elementz⊗a⊗b,yieldsz⊗b⊗a.If we assume that a given generalized bi-circular projectionP_λ,onX₁⊗ˇX₂, is associated with such an isometry T thenP_∗ =U⁻¹⊗Id_X2 ◦ P_λ ◦ U⊗Id_X2 is a projection (P_∗²=P_∗).

Therefore we have that (λ+ 1)(S−Id) = 0,and henceλ=−1 orS=Id. In either caseP_λ is the average of the Id with an isometric reflection (S²=Id).

Corollary 5.5. If X₁ is a complex Banach space with trivial centralizer and X₂ a complex Banach space with strictly convex dual then every generalized bi-circular projectionP onX₁⊗ˇX₂ is of the form

P₁⊗Id_X2, Id_X1⊗P₂, or Id_X1⊗X2+R

2 ,

where P_i are generalized bi-circular projections on X_i and R is an isometric re- flection onX₁⊗ˇX₂.

Proof. If the isometry associated withPis dyadic then corollary 5.2 applies. Other- wise Jarosz’s theorem asserts the existence of a Banach spaceZsuch thatX₁is iso- metrically isomorphic toZ⊗X₂where the isometry associated withP, denoted by R, is tensor conjugate to a reflection, henceR²=Id. SinceR²−(1+λ)R+λId = 0 thenλ=−1 andP= ^{IdX1 ˇ⊗X}₂ ^2+R.This completes the proof.

Corollary 5.6. 1. Every generalized bi-circular projection,P_λ with λ=−1, on L^p⊗ˆL^p is of the form

P₁⊗Id or Id⊗P₂,

whereP_i is a generalized bi-circular projection onL^p.

2. IfX andY define an ideal pair of Banach spaces, every generalized bi-circular projection,P_λ with λ=−1, onX⊗ˆY^∗ is of the form

P₁⊗Id_Y∗ or Id_X⊗P₂,

whereP₁ is a generalized bi-circular projection onX andP₂ is a generalized bi-circular projection onY^∗ .

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Fernanda Botelho, Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

e-mail:mbotelho@memphis.edu

James Jamison, Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

e-mail:jjamison@memphis.edu

Received: 26 February 2007 Revised: 30 May 2007