Let H be a complex Hilbert space; denote by Alg 𝒩 and 𝒞p(H) the atomic nest algebra associated with the atomic nest 𝒩 on H and the space of Schatten-p class operators on, H respectively. Let 𝒞p(H)∩Alg 𝒩 be the space of Schatten-p class operators in Alg 𝒩. When 1≤p<+∞ and p≠2, we give a complete characterization of nonlinear surjective isometries on 𝒞p(H)∩Alg 𝒩. If p=2, we also prove that a nonlinear surjective isometry on 𝒞2(H)∩Alg 𝒩 is the translation of an orthogonality preserving map.

1. Introduction

Let X and Y be normed spaces and let ϕ be a map from X to Y. We say that ϕ is a nonlinear isometry (or a distance preserving map) if ∥ϕ(x1)-ϕ(x2)∥=∥x1-x2∥ for every pair x1,x2 in X. In particular, ϕ is an isometry if ϕ is linear and distance preserving. The question of characterizing isometries between operator algebras is very important in studying geometric structure of operator algebras. Many authors pay their attention to such a problem (see [1–11] and their references). One of well-known results is due to Kadison, who states that every isometry for the operator norm from a unital C*-algebra onto another unital C*-algebra is a C*-isomorphism followed by left multiplication by a fixed unitary element (see [7]). Besides, of the operator norm, isometries for the Schatten-p norm also are studied extensively (see [1, 2, 5, 6, 8–11] and their references). Early in 1975, Arazy in [2] gave a characterization of isometries on the Schatten-p class (p≠2). In [1], Anoussis and Katavolos characterized isometries on the Schatten-p class in nest algebras and obtained the following theorem.

Theorem 1.

Let 𝒩and ℳ be two nests of projections on a Hilbert space H and J a fixed involution on H. Assume that 1≤p<+∞, p≠2. The surjective isometries Φ:𝒞p(H)∩
Alg 𝒩→𝒞p(H)∩
Alg ℳ have one of the following forms:
(1)Φ(A)=VAV*WorΦ(A)=VJA*JV*W,
where W is a unitary operator, and E↦VEV* is an order isomorphism of 𝒩 onto ℳ (E↦VJEJV* is an order isomorphism of 𝒩 onto ℳ⊥).

More generally, in recent years, many authors are devoted to characterizing distance preserving maps on operator algebras (see [3, 4, 12–15] and their references). In [14], Chan et al. showed that a nonlinear surjective isometry Φ for the unitarily invariant norm on n×m complex matrix algebras has one of the following forms.

There are unitary matrices U,V and a n×m matrix S such that Φ(A)=UAV+S or Φ(A)=UA¯V+S for each n×m matrix A.

If m=n and Φ has the form, there are unitary matrices U,V and a n×n matrix S such that Φ(A)=UAtV+S or Φ(A)=UA*V+S for each n×n matrix A.

If the unitarily invariant norm is a multiple of the Frobenius norm, that is, ∥A∥=λtr(AA*)1/2 for some λ>0, then the map A↦Φ(A)-S is a real orthogonal transformation with respect to the inner product (A,B)=Retr(AB*) for each n×m matrix A.

Recall that a norm ∥·∥ of operators is a unitary invariant norm if ∥UAV∥=∥A∥ for any unitary operators U,V. In [3, 4, 12, 13, 15], distance preserving maps on several kinds of operator algebras in the infinite dimensional case were characterized. Bai and Hou in [12] give a characterization of nonlinear numerical radius isometries on ℬ(H). Cui and Hou in [13] characterize nonlinear numerical radius isometries on atomic nest algebras and diagonal algebras. Hou and He in [15] give a characterization of nonlinear isometries on the Schatten-p class. A nature problem is how to characterize nonlinear isometries on the Schatten-p class in nest algebras. The main purpose of this paper is to give a complete characterization of nonlinear surjective isometries on the Schatten-p class (p≠2) in atomic nest algebras acting on Hilbert spaces (Theorem 2). Such a result generalizes the linear map assumption in Theorem 1 to the nonlinear case. Also, the problem on orthogonality of nonlinear surjective isometries for Hilbert-Schmidt norms is discussed (Theorem 3).

By the classical Mazur-Ulam theorem (see [10]) which states that every distance preserving surjective map sending 0 to 0 between normed spaces is real linear, we essentially deal with the real linear isometries (i.e., the distance preserving real linear maps). One can not expect that each real isometry has the same structure as the complex isometry. Applicable examples are found in [3] (Example 0.2, 0.3 in [3]).

Following the idea of [3], a key step in our approach is to show that the distance preserving maps on the Schatten-p class (p≠2) in nest algebras also preserve rank-one operators in both directions. This leads to a demand for characterizing rank-1 preserving additive maps between nest algebras. Related results had been obtained in [16].

Before embarking upon our results, it is convenient here to introduce some notations. Denote by ℝ or ℂ the real or complex field. For an operator A on H, we denote the range of A by ranA and the adjoint of A by A*. Let τ be an automorphism (or homomorphism) of 𝔽=ℝ or ℂ. If a map A on H satisfies A(x+y)=Ax+Ay and A(λx)=τ(λ)Ax for every x,y∈H and λ∈𝔽, then we say that A is τ-linear. If τ is a ring homomorphism, then we say that A is semilinear and in the case that 𝔽=ℂ and τ(λ)≡λ¯, we say A is conjugate linear. If U is a conjugate linear operator between Hilbert spaces and U*U=UU*=I, U is called conjugate unitary or antiunitary, where U* is the Hilbert space conjugate operator of U. Denote by 𝒦(H) and ℱ(H) the space of all compact operators and the space of all finite rank operators on the Hilbert space H. For any A∈𝒦(H), the trace of A, tr(A)=Σi〈Aei,ei〉, where {ei}(i∈I) is a normal orthogonal base in the Hilbert space H. Let |A|=(A*A)1/2; the Schatten-p norm of A is as follows:
(2)∥A∥p=tr(|A|p)1/p.
The Schatten-p class 𝒞p(H) is the set of all Schatten-p class operators, that is, all compact operators with the finite Schatten-p norm. If p=1, the set 𝒞1(H) is called the trace class. If p=∞, 𝒞∞(H)=𝒦(H). Recall that a nest on H is a chain 𝒩 of closed (under norm topology) subspaces of H containing {0} and H, which is closed under the formation of arbitrary closed linear span (denoted by ⋁) and intersection (denoted by ⋀). Alg 𝒩 denotes the associated nest algebra, which is the set of all operators T in ℬ(H) such that TN⊆N for every element N∈𝒩. If 𝒩 is a nest, 𝒩⊥={N⊥∣N∈𝒩} is a nest. If 𝒩≠{{0},H}, we say that 𝒩 is nontrivial. We denote Algℱ𝒩=Alg𝒩∩ℱ(H). For any N∈𝒩, let N-=⋁{M∈𝒩∣M⊂N}, N+=⋀{M∈𝒩∣N⊂M}, and N-⊥=(N-)⊥. 0-=0, H+=H. If N⊖N-=N∩(N-)⊥≠0, we say N⊖N- is an atom of 𝒩. A nest 𝒩 on H is said to be atomic if H is spanned by its atoms and to be maximal if 𝒩 is atomic and all its atoms are one-dimensional. The rank-one operator x⊗f∈ Alg 𝒩 if and only if there is an N∈𝒩 such that x∈N and f∈N-⊥. For each x∈H, Lx={x⊗f∣f∈H} and f∈H, Rf={x⊗f∣x∈H}. If 𝒩 is a nest and N∈𝒩, for x∈N, LxN={x⊗f∣f∈N-⊥}; for f∈N-⊥, RfN={x⊗f∣x∈N}. Assume that ℰ1(𝒩)=∪{N∈𝒩∣dimN-⊥>1}, ℰ2(𝒩)=∪{N-⊥∣N∈𝒩,dimN>1}, 𝒟1(𝒩)=∪{N∈𝒩∣N-≠H}, 𝒟2(𝒩)=∪{N-⊥∣N∈𝒩 and N≠0}, E1(𝒩)={N∣N∈𝒩,dimN-⊥>1}, and E2(𝒩)={N∈𝒩∣dimN>1}. If the nest is fixed, they are written briefly as ℰ1, ℰ2, 𝒟1, 𝒟2, E1, E2, respectively.

2. Main Results

In the following theorem, we give a characterization of nonlinear surjective isometries on 𝒞p(H)∩Alg𝒩, where 𝒩 is an atomic nest.

Theorem 2.

Let H be a complex Hilbert space, 𝒩 an atomic nest on H. Assume that 1≤p<+∞, p≠2, Φ:𝒞p(H)∩
Alg 𝒩→𝒞p(H)∩
Alg 𝒩 is a surjective map. Then Φ satisfies ∥Φ(A)-Φ(B)∥p=∥A-B∥p for all A,B∈𝒞p(H)∩
Alg 𝒩 if and only if one of the following holds true.

There exist an operator S∈𝒞p(H)∩
Alg 𝒩, a dimension preserving order isomorphism θ:𝒩→𝒩, and unitary operators U,V:H→H satisfying U(N)=θ(N) and V(N⊥)=θ(N)-⊥ for every N∈𝒩, such that
(3)Φ(A)=UAV+S,∀A∈𝒞p(H)∩
Alg 𝒩.

There exist an operator S∈𝒞p(H)∩
Alg 𝒩, a dimension preserving order isomorphism θ:𝒩→𝒩, and conjugate unitary operators U,V:H→H satisfying U(N)=θ(N) and V(N⊥)=θ(N)-⊥ for every N∈𝒩, such that
(4)Φ(A)=UAV+S,∀A∈𝒞p(H)∩
Alg 𝒩.

There exist an operator S∈𝒞p(H)∩
Alg 𝒩, a dimension preserving order isomorphism θ:𝒩⊥→𝒩⊥, and unitary operators U,V:H→H satisfying U(N⊥)=θ(N⊥) and V(N)=θ(N-⊥)-⊥ for every N∈𝒩, such that
(5)Φ(A)=UA*V+S,∀A∈𝒞p(H)∩
Alg 𝒩.

There exist an operator S∈𝒞p(H)∩
Alg 𝒩, a dimension preserving order isomorphism θ:𝒩⊥→𝒩⊥, and conjugate unitary operators U,V:H→H satisfying U(N⊥)=θ(N⊥) and V(N)=θ(N-⊥)-⊥ for every N∈𝒩, such that
(6)Φ(A)=UA*V+S,∀A∈𝒞p(H)∩
Alg 𝒩.

The problem on orthogonality of nonlinear surjective isometries for Hilbert-Schmidt norms is discussed in the following theorem.

Theorem 3.

Let H be a complex Hilbert space, 𝒩 an atomic nest on H. Assume that Φ:𝒞2(H)∩
Alg 𝒩→𝒞2(H)∩
Alg 𝒩 is a surjective map. Then Φ satisfies ∥Φ(A)-Φ(B)∥2=∥A-B∥2 for all A,B∈𝒞2(H)∩
Alg 𝒩 and then the map A↦Φ(A)+Φ(0) is a real linear and an orthogonal transformation on 𝒞2(H)∩
Alg 𝒩 with respect to the real inner product 〈A,B〉=
Retr
(AB*).

3. Proof of Main Results

To prove our main results, we need the following lemmas.

Lemma 4 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

For arbitrary A,B∈𝒞p(H) and 1≤p<+∞, p≠2, A*B=AB*=0 if and only if
(7)∥A-B∥pp+∥A+B∥pp=2(∥A∥pp+∥B∥pp).

In the following lemmas, we give a characterization of rank-oneness of operators by the relation of orthogonality between operators. Let {A}⊥={B∈𝒞p(H)∩Alg𝒩∖{0}:A*B=AB*=0} for arbitrary A∈𝒞p(H)∩Alg𝒩. The set {A}⊥ is maximal, if for arbitrary operator N∈𝒞p(H)∩Alg𝒩, {A}⊥⊆{N}⊥⇒{A}⊥={N}⊥.

Lemma 5 (see Lemma 3 in [<xref ref-type="bibr" rid="B1">1</xref>]).

For A=x⊗f∈𝒞p(H)∩
Alg 𝒩, then

∩{kerT:T∈{A}⊥}=[f] unless [x]=0+, in which case ∩{kerT:T∈{A}⊥}=[x,f];

∩{kerT*:T∈{A}⊥}=[x] unless [f]=H-⊥, in which case ∩{kerT*:T∈{A}⊥}=[x,f].

Lemma 6.

For any nonzero operator A∈𝒞p(H)∩
Alg 𝒩 with the atomic nest 𝒩, if the set {A}⊥ is maximal and nonempty, then rankA=1. Conversely, if rankA=1, and either 0+≠
ran A or H-⊥≠
ran
(A*), then the set {A}⊥ is maximal and nonempty.

Proof.

If {A}⊥ is maximal and nonempty, we show that rank A=1. If not, rank A≥2, then there are two nonzero vectors x1 and x2 such that Ax1⊥Ax2. Since the nest is atomic, let P=Ax1⊗A*x1=Ax1⊗x1A∈𝒞p(H)∩Alg𝒩 and one can find a vector y2⊥A*x1 such that Q=Ax2⊗y2∈𝒞p(H)∩Alg𝒩 (if necessary, interchanging x1 for x2). Now for any T∈{A}⊥, it follows from the definition of {A}⊥ that A*T=AT*=0. So P*T=A*x1⊗Ax1T=A*x1⊗x1A*T=0 and PT*=Ax1⊗A*x1T*=Ax1⊗TA*x1=0; it follows that T∈{P}⊥. So we have {P}⊥⊇{A}⊥. One can check P*Q=PQ*=0 but A*Q≠0. That is, Q∈{P}⊥ but is not in {A}⊥. It is a contradiction to the maximum of {A}⊥. So rank A=1.

If rank A=1, let A=x⊗f, and either 0+≠ranA or H-⊥≠ran(A*), and by Lemma 5, one of the following three cases happens.

Case 1. ∩{kerT:T∈{A}⊥}=[f] and ∩{kerT*:T∈{A}⊥}=[x].

Case 2. ∩{kerT:T∈{A}⊥}=[x,f] and ∩{kerT*:T∈{A}⊥}=[x].

Case 3. ∩{kerT:T∈{A}⊥}=[f] and ∩{kerT*:T∈{A}⊥}=[x,f].

If for N∈𝒞p(H)∩Alg𝒩, {A}⊥⊆{N}⊥, then either
(8)∩{kerT:T∈{N}⊥}⊆∩{kerT:T∈{A}⊥}⊆[f]
or
(9)∩{kerT*:T∈{N}⊥}⊆∩{kerT:T∈{A}⊥}⊆[x].
It follows that either ran(A*)⊆∩{kerT:T∈{N}⊥}⊆[f] or ranA⊆∩{kerT*:T∈{N}⊥}⊆[x]. It implies that rankN=1 and N,A are linearly dependent. By computation, then {A}⊥={N}⊥. So {A}⊥ is maximal and nonempty.

In the following lemma that is taken from [16], let X′ be the dual of a Banach space X. Let 𝒩 be a nest on X over real or complex field 𝔽. If dim 0+=1, ℰ2(𝒩)¯⊕[e0]=X, and if dim H-⊥=1, ℰ1(𝒩)¯⊕[f0]=X′.

Lemma 7 (see [<xref ref-type="bibr" rid="B4">16</xref>]).

Let 𝒩 and ℳ be two nests on Banach spaces X and Y over real or complex field 𝔽, respectively. Let Φ:
Alg
ℱ𝒩→
Alg
ℱℳ be a continuous surjective additive map. Then Φ preserves rank-1 operators in both directions if and only if one of the following is true.

There are linear or conjugate linear bounded bijective operators A:X→Y, C:X′→Y′, a dimension preserving order isomorphism θ:𝒩→ℳ, and vectors y0∈Y, g0∈Y′ such that A(N)=θ(N), C(N-⊥)=θ(N)-⊥ for every N∈𝒩, and for each rank-1 operator x⊗f∈
Alg
ℱ𝒩,
(10)Φ(x⊗f)={Ax⊗Cfifx∈ℰ1(𝒩)-,f∈ℰ2(𝒩)-,Ax⊗Cf+Imf(x)Ae0⊗g0ifx∈ℰ1(𝒩)-,f∉ℰ2(𝒩)-,Ax⊗Cf+Imf(x)y0⊗Cf0ifx∉ℰ1(𝒩)-,f∈ℰ2(𝒩)-.

There are linear or conjugate linear bounded bijective operators A:X′→Y, C:X→Y′, a dimension preserving order isomorphism θ:𝒩⊥→ℳ, and vectors y0∈Y, g0∈Y′ such that A(N-⊥)=θ(N-⊥), C(N)=θ(N-⊥)-⊥ for every N∈𝒩, and for each rank-1 operator x⊗f∈
Alg
ℱ𝒩,
(11)Φ(x⊗f)={Af⊗Cxifx∈ℰ1(𝒩),f∈ℰ2(𝒩),Af⊗Cx+Imf(x)y0⊗Ce0ifx∈ℰ1(𝒩),f∉ℰ2(𝒩),Af⊗Cx+Imf(x)Af0⊗g0ifx∉ℰ1(𝒩),f∈ℰ2(𝒩).

Moreover, in this case, X and Y are reflexive.
Lemma 8.

For any A,B∈𝒞2(H), the following are equivalent:

〈A,B〉=
Retr
(AB*)=0;

∥A+λB∥2≥∥A∥2 for any real number λ.

Proof.

(I)⇒(II) If 〈A,B〉=Retr(AB*)=0, for any real number λ,
(12)∥A∥22≤∥A∥22+∥λB∥22=tr(AA*)+λ2tr(BB*)+2λRetr(AB*)=∥A+λB∥22.

(II)⇒(I) Without loss of generality, assume that B≠0, and by (II), we have, for any real number λ,
(13)∥A∥22≤∥A+λB∥22=tr(|A+λB|)=∥A∥22+∥λB∥22+λtr(AB*)+λtr(BA*);
that is, λ2∥B∥22≥-λtr(AB*)-λtr(BA*). So λ∥B∥22≥-Retr(AB*). It follows from arbitrariness of λ that Retr(AB*)=0. We complete the proof.

Proof of Theorem <xref ref-type="statement" rid="thm2.1">2</xref>.

Checking the “if” part is straightforward, so we will only deal with the “only if” part.

Let Ψ(A)=Φ(A)-Φ(0) for any A∈𝒞p(H)∩Alg𝒩; then Ψ(0)=0, and ∥Ψ(A)-Ψ(B)∥p=∥A-B∥p for any A,B∈𝒞p(H)∩Alg𝒩. By the Mazur-Ulam theorem (see [10]), we have that Ψ is an additive map. Furthermore, we have that ∥Ψ(A)∥p=∥A∥p and ∥Ψ(A)±Ψ(B)∥p=∥A±B∥p for any A,B∈𝒞p(H)∩Alg𝒩. By Lemma 4, Ψ satisfies that A*B=AB*=0⇔Ψ(A)*Ψ(B)=Ψ(A)Ψ(B)*=0 for all A,B∈𝒞p(H)∩Alg𝒩.

Next we show that Ψ preserves rank-one operators in both directions. For any rank-one operator A=x⊗f, by the above discussion, Ψ({A}⊥)={Ψ(A)}⊥. By Lemma 6, if either 0+≠ranA or H-⊥≠ran(A*), then Ψ(A) has rank one. Ψ-1 has the same property as Ψ, and Ψ-1 preserves rank-one operators. So Ψ preserves rank-one operators in both directions. If both 0+=ranA and H-⊥=ran(A*), take rank-one operator An=x⊗((1/n)x+f); then An→A(n→∞). So we have Ψ(An)→Ψ(A). Since Ψ(An) has rank one, then Ψ(A) has rank one. As Ψ-1 has the same property as Ψ, so Ψ preserves rank-one operators in both directions.

Ψ preserves rank-one operators in both directions; then Ψ has the form in Lemma 7. In the case of complex Hilbert space, we have that one of the following is true.

There are linear or conjugate linear bounded bijective operators A,C:H→H, vectors y0,g0∈H, and a dimension preserving order isomorphism θ:𝒩→𝒩 such that A(N)=θ(N), C(N-⊥)=θ(N)-⊥ for every N∈𝒩, such that
(14)Φ(x⊗y)={Ax⊗Cyifx∈ℰ1¯,y∈ℰ2¯,Ax⊗Cy+Im〈x,y〉Ae0⊗g0ifx∈ℰ1¯,y∉ℰ2¯,Ax⊗Cy+Im〈x,y〉y0⊗Cf0ifx∉ℰ1¯,y∈ℰ2¯.

There are linear or conjugate linear bounded bijective operators A,C:H→H, vectors y0,g0∈H, and a dimension preserving order isomorphism θ:𝒩⊥→𝒩 such that A(N-⊥)=θ(N-⊥), C(N)=θ(N-⊥)-⊥ for every N∈𝒩, such that
(15)Φ(x⊗y)={Ay⊗Cxifx∈ℰ1¯,y∈ℰ2¯,Ay⊗Cx+Im〈x,y〉y0⊗Ce0ifx∈ℰ1¯,y∉ℰ2¯,Ay⊗Cx+Im〈x,y〉Af0⊗g0ifx∉ℰ1¯,y∈ℰ2¯.

One can note that ∥T∥=∥T∥p for all rank-one operator T, so is the same to the proof of Lemma 4.11 in [3], A,C can be chosen as unitary or conjugate unitary operators; denote A,C by U,W.

If case (1) occurs, next we claim Ue0⊗g0=0, and y0⊗Wf0=0. For case (2), similarly, we can show that y0⊗We0=0 and Uf0⊗g0=0. Assume that (1) occurs and Ψ has the second form, in fact dim0+=1, ℰ2¯=[e0]⊥. Just like the discussion in Lemma 4.11 in [3], we have g0=e0, We0=e0. Assume on the contrary that Ue0⊗g0≠0. Let A=ie0⊗e0, Ψ(A)=Ψ(ie0⊗e0)=iUe0⊗We0+Ue0⊗e0; then ∥A∥p=∥ie0⊗e0∥p=∥ie0∥∥e0∥=1. By a computation, ∥Ψ(A)∥p=∥iUe0⊗We0+Ue0⊗e0∥p=∥ie0⊗e0+e0⊗W*e0∥p=∥ie0⊗e0+e0⊗e0∥p=∥(ie0+e0)⊗e0∥p≠1. So ∥Ψ(A)∥p≠∥A∥p, a contradiction. So Ue0⊗g0=0. If Ψ has the third form, in this case dim H-⊥=1, ℰ1¯=[f0]⊥. Just like the discussion in Lemma 4.11 in [3] again, y0=f0=Ue0. Assume on the contrary that y0⊗Wf0≠0; let A=if0⊗f0, similar to the above discussion, and we get a contradiction again.

Every finite rank operator can be written as a sum of rank-one operators in the nest algebra, and the set of finite rank operators is dense in 𝒞p(H)∩Alg𝒩, so by addition of Ψ, we have that Theorem 2 holds true by replacing W* by V.

Proof of Theorem <xref ref-type="statement" rid="thm2.2">3</xref>.

Let Ψ(A)=Φ(A)-Φ(0) for any A∈𝒞2(H); then Ψ(0)=0, and ∥Ψ(A)-Ψ(B)∥2=∥A-B∥2 for A,B∈𝒞2(H)∩
Alg𝒩. By the Mazur-Ulam theorem (see [10]), Ψ is real linear.

By real linearity of Ψ, we have that ∥A+λB∥22=∥Ψ(A)+λΨ(B)∥22 for any real number λ. By Lemma 8, Ψ is real linear and the map Ψ preserves orthogonality with respect to 〈A,B〉=
tr(AB*). We complete the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported partially by National Science Foundation of China (11201329, 11171249).

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