Exact and approximate operator parallelism

0.2 MB·

by  Mohammad Sal Moslehian

EXACT AND APPROXIMATE OPERATOR PARALLELISM
4 MOHAMMAD SAL MOSLEHIANAND ALIZAMANI
1
0
2 Abstract. Extendingthenotionofparallelismweintroducetheconceptofapproximateparal-

y lelism innormedspaces andthensubstantiallyrestrict ourselvestothesettingof Hilbertspace
a
operators endowed with the operator norm. We present several characterizations of the exact
M
and approximate operator parallelism in the algebra B(H) of bounded linear operators acting

8 on a Hilbert space H. Among other things, we investigate the relationship between approx-
1
imate parallelism and norm of inner derivations on B(H). We also characterize the parallel

] elements of a C∗-algebra by using states. Finally we utilize the linking algebra to give some
A
equivalenceassertions regarding parallel elements in a Hilbert C∗-module.
F
.
h
t
a
m 1. Introduction and preliminaries
[
Let A be a C∗-algebra. An element a A is called positive (we write a 0) if a = b∗b for
2 ∈ ≥
some b A. If a A is positive, then exists a unique positive element b A such that a = b2.
v
∈ ∈ ∈
9 Such an element b is called the positive square root of a. A linear functional ϕ over A of norm
6
1 one is called state if ϕ(a) 0 for any positive element a A. By S(A) we denote the set of all
3 ≥ ∈
. states of A.
1
0 Throughout the paper, K(H ) and B(H ) denote the C∗-algebra of all compact operators and
4
the C∗-algebra of all bounded linear operators on a complex Hilbert space H endowed with an
1
v: inner product (. .), respectively. We stand I for the identity operator on H . Furthermore,
|
Xi For ξ,η H , the rank one operator ξ η on H is defined by (ξ η)(ζ) = (ζ η)ξ. Note that
∈ ⊗ ⊗ |
r by the Gelfand–Naimark theorem we can regard A as a C∗-subalgebra of B(H ) for a complex
a
Hilbert space H . More details can be found e.g. in [5, 15].
The notion of Hilbert C∗-module is a natural generalization of that of Hilbert space arising
under replacement of the field of scalars C by a C∗-algebra. This concept plays a significant
role in the theory of operator algebra and K-theory; see [12]. Let A be a C∗-algebra. An inner
product A-module is a complex linear space X which is a right A-module with a compatible
scalar multiplication (i.e., µ(xa) = (µx)a = x(µa) for all x X,a A,µ C) and equipped
∈ ∈ ∈
with an A-valued inner product , :X X A satisfying
h· ·i × −→
(i) x,αy +βz =α x,y +β x,z ,
h i h i h i
(ii) x,ya = x,y a,
h i h i
2010 Mathematics Subject Classification. 47A30, 46L05, 46L08, 47B47, 15A60.
Key words and phrases. C∗-algebra; approximate parallelism; operator parallelism; Hilbert C∗-module.
1
2 M.S.MOSLEHIAN,A.ZAMANI
(iii) x,y ∗ = y,x ,
h i h i
(iv) x,x 0 and x,x = 0 if and only if x = 0,
h i ≥ h i
for all x,y,z X,a A,α,β C. For an inner product A-module X the Cauchy–Schwarz
∈ ∈ ∈
inequality holds (see [7] and references therein):
x,y 2 x,x y,y (x,y X).
kh ik ≤ kh ikkh ik ∈
Consequently, x = x,x 12 definesanormonX. IfX withrespecttothisnormiscomplete,
k k kh ik
then it is called a Hilbert A-module, or a Hilbert C∗-module over A. Complex Hilbert spaces
are Hilbert C-modules. Any C∗-algebra A can be regarded as a Hilbert C∗-module over itself
via a,b := a∗b. For every x X the positive square root of x,x is denoted by x . If ϕ be a
h i ∈ h i | |
state over A, we have the following useful version of the Cauchy-Schwarz inequality:
ϕ( y,x )ϕ( x,y ) = ϕ( x,y )2 ϕ( x,x )ϕ( y,y )
h i h i | h i | ≤ h i h i
for all x,y X.
∈
LetX andY betwoHilbertA-modules. AmappingT : X Y iscalled adjointable ifthere
−→
exists a mapping S : Y X such that Tx,y = x,Sy for all x X,y Y . The unique
−→ h i h i ∈ ∈
mapping S is denoted by T∗ and is called the adjoint of T. It is easy to see that T must be
boundedlinearA-modulemapping. ThespaceB(X,Y )ofalladjointablemapsbetweenHilbert
A-modulesX andY isaBanachspacewhileB(X) := B(X,X)isaC∗-algebra. ByK(X,Y )
we denote the closed linear subspace of B(X,Y ) spanned by θ : x X,y Y , where
y,x
{ ∈ ∈ }
θ is defined by θ (z) = y x,z . Elements of K(X,Y ) are often referred to as “compact”
y,x y,x
h i
operators. We write K(X) for K(X,X).
Any Hilbert A-module can be embedded into a certain C∗-algebra. To see this, let X A
⊕
be the direct sum of the Hilbert A-modules X and A equipped with the A-inner product
(x,a),(y,b) = x,y + a∗b, for every x,y X,a,b A. Each x X induces the maps
h i h i ∈ ∈ ∈
r B(A,X) and l B(X,A) given by r (a) = xa and l (y) = x,y , respectively, such that
x x x x
∈ ∈ h i
r∗ = l . Themapx r is an isometric linear isomorphismof X toK(A,X) andx l is an
x x x x
7→ 7→
isometric conjugate linear isomorphism of X to K(X,A). Further, every a A induces the
∈
map T K(A) given by T (b) = ab. The map a T defines an isomorphism of C∗-algebras
a a a
∈ 7→
A and K(A). Set
K(A) K(X,A) T l
L(X) = = a y : a A, x,y X, T K(X)
"K(A,X) K(X) # ("rx T# ∈ ∈ ∈ )
Then L(X) is a C∗-subalgebra of K(X A), called the linking algebra of X. Clearly
⊕
0 0 A 0 0 0
X , A , K(X) .
≃ "X 0# ≃" 0 0# ≃ "0 K(X)#
EXACT AND APPROXIMATE OPERATOR PARALLELISM 3
Furthermore, x,y of X becomes the product l r in L(X) and the module multiplication of
x y
h i
X becomes a part of the internal multiplication of L(X). We refer the reader to [11, 16] for
more information on Hilbert C∗-modules and linking algebras.
Following Seddik [19] we introduce a notion of parallelism in normed spaces in Section
2. Inspired by the approximate Birkhoff–James orthogonality (ε-orthogonality) introduced by
Dragomir [6] and a variant of ε-orthogonality given by Chmielin´ski [4] which has been investi-
gated by Iliˇsevi´c and Turnˇsek [10] in the setting of Hilbert C∗-modules, we introduce a notion
of approximate parallelism (ε-parallelism).
Inthenextsections,wesubstantiallyrestrictourselvestothesettingofHilbertspaceoperators
equipped with the operator norm. In section 3, we present several characterizations of the exact
and approximate operator parallelism in the algebra B(H ) of bounded linear operators acting
onaHilbertspaceH . Amongotherthings,weinvestigate therelationshipbetweenapproximate
parallelism and norm of inner derivations on B(H ). In Section 4, we characterize the parallel
elements of a C∗-algebra by using states and utilize the linking algebra to give some equivalence
assertions regarding parallel elements in a Hilbert C∗-module.
2. Parallelism in normed spaces
We start our work with the following definition of parallelism in normed spaces.
Definition 2.1. Let V bea normed space. Thevector x V is exact parallel or simply parallel
∈
to y V , denoted by x y (see [19]), if
∈ k
x+λy = x + y , for some λ T= α C : α = 1 . (2.1)
k k k k k k ∈ { ∈ | | }
Notice that the parallelism is a symmetric relation. It is easy to see that if x,y are linearly
dependent, then x y. The converse is however not true, in general.
k
Example 2.2. Let us consider the space (R2, . ) where (x ,x ) = max x , x for all
1 2 1 2
|||||| ||| ||| {| | | |}
(x ,x ) R2. Let x = (1,0),y = (1,1) and λ = 1. Then x,y are linearly independent and
1 2
∈
x+λy = (2,1) = 2 = x + y , i.e., x y.
||| ||| ||| ||| ||| ||| ||| ||| k
An operator T on a separable complex Hilbert space is said to be in the Schatten p-class
1
p (1 p < ), if tr(T p) < . The Schatten p-norm of T is defined by T p = (tr(T p))p.
C ≤ ∞ | | ∞ k k | |
For 1 < p 2 and 1 + 1 = 1, the Clarkson inequality for T,S asserts that
≤ p q ∈ Cp
q
T +S q + T S q 2( T p + S p)p,
p p p p
k k k − k ≤ k k k k
which can be found in [13].
Theorem 2.3. Let T,S with 1 < p 2 and 1 + 1 = 1. The following statements are
∈ Cp ≤ p q
equivalent:
4 M.S.MOSLEHIAN,A.ZAMANI
(i) T,S are linearly dependent;
(ii) T S.
k
Proof. Obviously, (i)= (ii).
⇒
Suppose (ii) holds. Therefore T + λS = T + S for some λ T. Without loss of
p p p
k k k k k k ∈
generality we may assume that T S . We have
p p
k k ≤ k k
λ T λ T T
p p p
2 T = T + k k S T + k k S = T +λS λ 1 k k S
p p
k k k k S ≥ S − − S
(cid:13) k kp (cid:13)p (cid:13) k kp (cid:13)p (cid:13) (cid:18) k kp(cid:19) (cid:13)p
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) T (cid:13) (cid:13) (cid:13) T (cid:13)
p p
T +λS(cid:13) 1 (cid:13) k k(cid:13) S = T(cid:13) +(cid:13)S 1 k k S =(cid:13)2 T ,
p p p p p p
≥ k k − − S k k k k k k − − S k k k k
p p
(cid:18) k k (cid:19) (cid:18) k k (cid:19)
so T + λkTkpS = 2 T . Hence by Clarkson inequality we get
kSkp p k kp
(cid:13) (cid:13)
(cid:13) (cid:13) λ T q λ T q λ T q
(cid:13) 2q T q +(cid:13) T k kpB = T + k kpS + T k kpS
k kp − S S − S
(cid:13) k kp (cid:13)p (cid:13) k kp (cid:13)p (cid:13) k kp (cid:13)p
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
q
(cid:13)(cid:13) (cid:13)(cid:13) 2(cid:13)(cid:13) T p + λk(cid:13)(cid:13)Tkp(cid:13)(cid:13)S p p = 21+pq(cid:13)(cid:13) T q = 2q T q,
≤ k kp S k kp k kp
(cid:13) k kp (cid:13)p!
(cid:13) (cid:13)
wherefrom we get T λkTkpS q = 0. Hence T =(cid:13)(cid:13) λkTkpS,(cid:13)(cid:13)which gives (i). (cid:3)
− kSkp p kSkp
(cid:13) (cid:13)
The following im(cid:13)portant exam(cid:13)ple is the motivation for further discussion.
(cid:13) (cid:13)
Example 2.4. If τ ,τ are positive linear functionals on a C∗-algebra A. Then for λ = 1 T,
1 2
∈
by [15, Corollary 3.3.5] we have τ +λτ = τ + τ . So τ τ .
1 2 1 2 1 2
k k k k k k k
Example 2.5. Suppose that τ is a self-adjoint bounded linear functional on a C∗-algebra.
By Jordan Decomposition Theorem [15, Theorem 3.3.10], there exist positive linear functionals
τ ,τ suchthatτ = τ τ and τ = τ + τ . Thusforλ = 1 Twehave τ +λτ =
+ − + − + − + −
− k k k k k k − ∈ k k
τ + τ . Hence τ τ .
+ − + −
k k k k k
For every ε [0,1), the following notion of the approximate Birkhoff–James orthogonality
∈
(ε-orthogonality) was introduced by Dragomir [6] as
x ε y x+λy (1 ε) x (λ C).
⊥ ⇐⇒ k k ≥ − k k ∈
In addition, an alternative definition of ε-orthogonality was given by Chmielin´ski [4]. These
facts motivate us to give the following definition of approximate parallelism (ε-parallelism) in
the setting of normed spaces.
Definition 2.6. Twoelements xandy inaanormedspaceareapproximate parallel (ε-parallel),
denoted by x ε y, if
k
inf x+µy : µ C ε x . (2.2)
{k k ∈ }≤ k k
EXACT AND APPROXIMATE OPERATOR PARALLELISM 5
It is remarkable that the relation ε-parallelism for ε = 0 is the same as the exact parallelism.
Proposition 2.7. In a normed space, the 0-parallelism is the same as exact parallelism.
Proof. Letusassumethatx = 0andchoosesequence µ ofvectorsinCsuchthatlim x+
n n→∞
6 { } k
µ y = 0. It follows from µ y x + µ y + x that the sequence µ is bounded.
n n n n
k | |k k ≤ k k k k { }
Therefore there exists a subsequence µ which is convergent to a number µ . Since x = 0
{ kn} 0 6
and lim x+µ y = 0, we conclude that µ = 0 as well as x+µ y = 0, or equivalently,
n→∞ n 0 0
k k 6 k k
x = µ y. Put λ = |µ0| T. Then
− 0 − µ0 ∈
µ
0
x+λy = µ y | |y = (µ +1) y = µ y + y = x + y ,
0 0 0
k k − − µ | | k k k− k k k k k k k
(cid:13) 0 (cid:13)
(cid:13) (cid:13)
whence x+λy = x(cid:13) + y for som(cid:13)e λ T, i.e., x y.
(cid:13) (cid:13)
k k k k k k ∈ k
(cid:3)
From now on we deal merely with the space B(H ) endowed with the operator norm.
3. Operator parallelism
In the present section, we discuss the exact and approximate operator parallelism. These
notions can be defined by the same formulas as (2.1) and (2.2) in normed spaces. Thus
T T T +λT = T + T
1 2 1 2 1 2
k ⇔ k k k k k k
for some λ T. The following example shows that the concept of operator parallelism is
∈
important.
Example 3.1. Suppose that T is a compact self-adjoint operator on a Hilbert space H . Then
either T or T is an eigenvalue of T. We may assume that T = 1 is an eigenvalue of T.
k k −k k k k
Therefore there exists a nonzero vector x H such that Tx = x. Hence 2 x = (T +I)x
∈ k k k k ≤
T +I x 2 x . So we get T +I = 2 = T + I . Thus T I and T fulfils the Daugavet
k kk k ≤ k k k k k k k k k
equation T +I = T +1; see [20]. This shows that the Daugavet equation is closely related
k k k k
to the notion of parallelism.
In the following proposition we state some basic properties of operator parallelism.
Proposition 3.2. Let T ,T B(H ). The following statements are equivalent:
1 2
∈
(i) T T ;
1 2
k
(ii) T∗ T∗;
1 k 2
(iii) αT βT (α,β Rr 0 );
1 2
k ∈ { }
(iv) γT γT (γ Cr 0 ).
1 2
k ∈ { }
6 M.S.MOSLEHIAN,A.ZAMANI
Proof. The equivalences (i) (ii) (iv) immediately follow from the definition of operator par-
⇔ ⇔
allelism.
(i)= (iii) Suppose that α,β Rr 0 and T T . Hence T +λT = T + T for some
1 2 1 2 1 2
⇒ ∈ { } k k k k k k k
λ T. We can assume that α β > 0. We therefore have
∈ ≥
αT + βT αT +λ(βT ) = α(T +λT ) (α β)(λT )
1 2 1 2 1 2 2
k k k k ≥ k k k − − k
α(T +λT ) (α β)λT = α T +λT (α β) T
1 2 2 1 2 2
≥ k k−k − k k k− − k k
= α( T + T ) (α β) T = αT + βT ,
1 2 2 1 2
k k k k − − k k k k k k
whence αT +λ(βT ) = αT + βT for some λ T. So αT βT .
1 2 1 2 1 2
k k k k k k ∈ k
(iii)= (i) is obvious. (cid:3)
⇒
In what follows σ(T) and r(T) stand for the spectrum and spectral radius of an arbitrary
element T B(H ), respectively. In the following theorem we shall characterize the operator
∈
parallelism.
Theorem 3.3. Let T ,T B(H ). Then the following statements are equivalent:
1 2
∈
(i) T T ;
1 2
k
(ii) There exist a sequence of unit vectors ξ in H and λ T such that
n
{ } ∈
lim (T ξ T ξ ) = λ T T ;
1 n 2 n 1 2
n→∞ | k kk k
(iii) r(T∗T ) = T∗T = T T ;
2 1 k 2 1k k 1kk 2k
(iv) T∗T T∗T and T∗T = T T ;
1 1 k 1 2 k 1 2k k 1kk 2k
(v) T∗(T +λT ) = T ( T + T ) for some λ T.
k 1 1 2 k k 1k k 1k k 2k ∈
Proof. (i) (ii) Let T T . Then T +λT = T + T for some λ T. Since
1 2 1 2 1 2
⇔ k k k k k k k ∈
sup T ξ+λT ξ : ξ H , ξ = 1 = T +λT = T + T ,
1 2 1 2 1 2
{k k ∈ k k } k k k k k k
thereexistsasequenceofunitvectors ξ inH suchthatlim T ξ +λT ξ = T + T .
n n→∞ 1 n 2 n 1 2
{ } k k k k k k
We have
T 2+2 T T + T 2 = lim T ξ +λT ξ 2
1 1 2 2 1 n 2 n
k k k kk k k k n→∞k k
= lim T ξ 2+(T ξ λT ξ )+(λT ξ T ξ )+ T ξ 2
1 n 1 n 2 n 2 n 1 n 2 n
n→∞ k k | | k k
T 2(cid:2)+2 lim (T ξ λT ξ ) + T 2 (cid:3)
1 1 n 2 n 2
≤ k k n→∞| | | k k
T 2+2 T T + T 2.
1 1 2 2
≤ k k k kk k k k
Therefore lim (T ξ λT ξ ) = T T , or equivalently,
n→∞ 1 n 2 n 1 2
| | k kk k
lim (T ξ T ξ ) = λ T T .
1 n 2 n 1 2
n→∞ | k kk k
EXACT AND APPROXIMATE OPERATOR PARALLELISM 7
To prove the converse, suppose that there exist a sequence of unit vectors ξ in H and λ T
n
{ } ∈
such that lim (T ξ T ξ ) = λ T T . It follows from
n→∞ 1 n 2 n 1 2
| k kk k
T T = lim (T ξ T ξ ) lim T ξ T T T ,
1 2 1 n 2 n 1 n 2 1 2
k kk k n→∞| | | ≤ n→∞k kk k≤ k kk k
that lim T ξ = T and by using a similar argument, lim T ξ = T . So that
n→∞ 1 n 1 n→∞ 2 n 2
k k k k k k k k
lim Re(T ξ λT ξ )= lim (T ξ λT ξ ) = T T ,
1 n 2 n 1 n 2 n 1 2
n→∞ | n→∞ | k kk k
whence we reach
1
T + T T +λT lim T ξ +λT ξ 2 2
1 2 1 2 1 n 2 n
k k k k ≥ k k ≥ n→∞k k
(cid:16) (cid:17) 1
= lim T ξ 2+2Re(T ξ λT ξ )+ T ξ 2 2
1 n 1 n 2 n 2
n→∞ k k | k k
(cid:16) (cid:2) 1 (cid:3)(cid:17)
= T 2+2 T T + T 2 2 = T + T .
1 1 2 2 1 2
k k k kk k k k k k k k
(cid:0) (cid:1)
Thus T +λT = T + T , so T T .
1 2 1 2 1 2
k k k k k k k
(ii) (iii) Let ξ be a sequence of unit vectors in H satisfying lim (T ξ T ξ ) =
n n→∞ 1 n 2 n
⇔ { } |
λ T T , for some λ T. By the equivalence (i) (ii) we have T T . Hence
1 2 1 2
k kk k ∈ ⇔ k
( T + T )2 = T +λT 2 = (T +λT )∗(T +λT )
1 2 1 2 1 2 1 2
k k k k k k k k
= T∗T +λT∗T +λT∗T +T∗T
k 1 1 1 2 2 1 2 2k
T∗T + λT∗T + λT∗T + T∗T
≤ k 1 1k k 1 2k k 2 1k k 2 2k
= T 2 +2 T T + T 2
1 1 2 2
k k k kk k k k
= ( T + T )2,
1 2
k k k k
so T∗T = T T .
k 1 2k k 1kk 2k
Since
T T = lim (T ξ T ξ ) = lim (T∗T ξ ξ ) lim T∗T ξ T∗ T = T T ,
k 1kk 2k n→∞| 1 n | 2 n | n→∞| 2 1 n | n | ≤ n→∞k 2 1 nk≤ k 2kk 1k k 1kk 2k
we have lim T∗T ξ = T T . Next observe that
n→∞k 2 1 nk k 1kk 2k
(T∗T λ T T I)ξ 2 = T∗T ξ 2 λ T T (T ξ T ξ )
k 2 1− k 1kk 2k nk k 2 1 nk − k 1kk 2k 1 n | 2 n
λ T T (T ξ T ξ )+ T 2 T 2.
1 2 2 n 1 n 1 2
− k kk k | k k k k
Therefore lim (T∗T λ T T I)ξ = 0. Thus r(T∗T ) = T T = T∗T .
n→∞k 2 1− k 1kk 2k nk 2 1 k 1kk 2k k 2 1k
The proof of the converse follows from the spectral inclusion theorem [9, Theorem 1.2-1] that
σ(T∗T ) (T∗T ξ ξ) : ξ H , ξ = 1 , where the bar denotes the closure.
2 1 ⊆ { 2 1 | ∈ k k }
(ii)= (iv) Let ξ be a sequence of unit vectors in H which satisfies
n
⇒ { }
lim (T ξ T ξ ) = λ T T ,
1 n 2 n 1 2
n→∞ | k kk k
8 M.S.MOSLEHIAN,A.ZAMANI
forsomeλ T. Asintheproofsoftheimplications(ii)= (i)and(ii)= (iii),weget T +λT =
1 2
∈ ⇒ ⇒ k k
T + T and T∗T = T T . By [15, Theorem 3.3.6] there is a state ϕ over B(H ) such
k 1k k 2k k 1 2k k 1kk 2k
that
ϕ((T +λT )∗(T +λT )) = (T +λT )∗(T +λT ) = T +λT 2 = ( T + T )2.
1 2 1 2 1 2 1 2 1 2 1 2
k k k k k k k k
Thus
( T + T )2 = ϕ(T∗T +λT∗T +λT∗T +T∗T )
k 1k k 2k 1 1 1 2 2 1 2 2
= ϕ(T∗T )+ϕ(λT∗T +λT∗T )+ϕ(T∗T )
1 1 1 2 2 1 2 2
T∗T + λT∗T +λT∗T + T∗T
≤ k 1 1k k 1 2 2 1k k 2 2k
= T∗T + T∗T + T∗T + T∗T
k 1 1k k 1 2k k 2 1k k 2 2k
T 2+2 T T + T 2
1 1 2 2
≤ k k k kk k k k
= ( T + T )2.
1 2
k k k k
Therefore ϕ(T∗T )= T∗T and ϕ(λT∗T )= T∗T . Hence
1 1 k 1 1k 1 2 k 1 2k
T∗T + T∗T = ϕ(T∗T +λT∗T ) T∗T +λT∗T T∗T + T∗T .
k 1 1k k 1 2k 1 1 1 2 ≤ k 1 1 1 2k ≤ k 1 1k k 1 2k
Therefore T∗T +λT∗T = T∗T + T∗T for some λ T. Thus T∗T λT∗T .
k 1 1 1 2k k 1 1k k 1 2k ∈ 1 1 k 1 2
(iv)= (v) This implication is trivial.
⇒
(v)= (i) Let T∗(T +λT ) = T ( T + T ) for some λ T. Then we have
⇒ k 1 1 2 k k 1k k 1k k 2k ∈
T ( T + T ) T∗ T +λT T∗(T +λT ) = T ( T + T ).
k 1k k 1k k 2k ≥ k 1kk 1 2k ≥ k 1 1 2 k k 1k k 1k k 2k
Thus T +λT = T + T , or equivalently, T T . (cid:3)
1 2 1 2 1 2
k k k k k k k
As an immediate consequence of Theorem 3.3, we get a characterization of operator paral-
lelism.
Corollary 3.4. Let T ,T B(H ). Then the following statements are equivalent:
1 2
∈
(i) T T ;
1 2
k
(ii) T ∗T T ∗T and T ∗T = T T (1 i = j 2);
i i j i j i j i
k k k k kk k ≤ 6 ≤
(iii) T T ∗ T T ∗ and T T ∗ = T T (1 i = j 2).
i i i j i j i j
k k k k kk k ≤ 6 ≤
Corollary 3.5. Let T ,T B(H ). Then the following statements are equivalent:
1 2
∈
(i) T T ;
1 2
k
(ii) r(T∗T ) = T∗T = T T ;
2 1 k 2 1k k 1kk 2k
(iii) r(T T∗) = T T∗ = T T .
1 2 k 1 2k k 1kk 2k
We need the next lemma for studying approximate parallelism.
EXACT AND APPROXIMATE OPERATOR PARALLELISM 9
Lemma 3.6. [1, Proposition 2.1] Let T ,T B(H ). Then
1 2
∈
inf T +µT 2 : µ C = sup M (ξ) : ξ C, ξ = 1 ,
{k 1 2k ∈ } { T1,T2 ∈ k k }
where
T ξ 2 |(T1ξ|T2ξ)|2 if T ξ = 0
M (ξ) = k 1 k − kT2ξk2 2 6
T1,T2  T ξ 2 if T ξ = 0.
k 1 k 2
In the following proposition we present a characterization of the operator ε-parallelism ε.
k
Recall that for T ,T B(H ) and ε [0,1), T ε T if inf T +µT : µ C ε T .
1 2 1 2 1 2 1
∈ ∈ k {k k ∈ } ≤ k k
Theorem 3.7. Let T ,T B(H ) and ε [0,1). Then the following statements are equivalent:
1 2
∈ ∈
(i) T ε T ;
1 2
k
(ii) T∗ ε T∗;
1 k 2
(iii) αT ε βT , (α,β Cr 0 ).
1 2
k ∈ { }
(iv) sup (T ξ η) : ξ = η = 1, (T ξ η) = 0 ε T .
1 2 1
{| | | k k k k | } ≤ k k
Moreover, each of the above conditions implies
(v) (T ξ T ξ)2 T ξ 2 T ξ 2 ε2 T 2 T 2, (ξ H , ξ = 1).
1 2 1 2 1 2
| | | ≥ k k k k − k k k k ∈ k k
Proof. (i) (ii) is obvious.
⇔
(i) (iii) Let T ε T and α,β Cr 0 . Then
1 2
⇔ k ∈ { }
µβ
inf αT +µ(βT ) : µ C = α inf T + T : µ C
1 2 1 2
{k k ∈ } | | {k α k ∈ }
α inf T +νT : ν C αε T = ε αT .
1 2 1 1
≤ | | {k k ∈ } ≤ | | k k k k
Therefore, αT ε βT . The converse is obvious.
1 2
k
(i) (iv) Bhatia and Sˇemrl [3, Remark 3.1] proved that
⇔
inf T +µT : µ C = sup (T ξ η) : ξ = η = 1, (T ξ η) = 0
1 2 1 2
{k k ∈ } {| | | k k k k | }
Thus the required equivalence follows from the above equality.
Now suppose that T ε T . Hence inf T +µT : µ C ε T . For any ξ H with
1 2 1 2 1
k {k k ∈ } ≤ k k ∈
ξ = 1, by Lemma 3.6, we therefore get
k k
T ξ 2 T ξ 2 (T ξ T ξ)2 T ξ 2inf T +µT 2 : µ C
1 2 1 2 2 1 2
k k k k −| | | ≤ k k {k k ∈ }
T ξ 2ε2 T 2 ε2 T 2 T 2 ξ 2 = ε2 T 2 T 2.
2 1 1 2 1 2
≤ k k k k ≤ k k k k k k k k k k
(cid:3)
In the following result we establish some equivalence statements to the approximate paral-
lelism for elements of a Hilbert space. We use some techniques of [14, Corollary 2.7] to prove
this corollary.
10 M.S.MOSLEHIAN,A.ZAMANI
Corollary 3.8. Let ξ,η H . Then for any ε [0,1) the following statements are equivalent:
∈ ∈
(i) ξ ε η;
k
(ii) sup (ξ ζ) : ζ H , ζ = 1,(η ζ)= 0 ε ξ ;
{| | | ∈ k k | } ≤ k k
(iii) (ξ η) √1 ε2 ξ η ;
| | | ≥ − k kk k
(v) η 2ξ (ξ η)η ε ξ η 2.
k k − | ≤ k kk k
(cid:13) (cid:13)
Proo(cid:13)f. Let ψ be a un(cid:13)it vector of H and set T = ξ ψ and T = η ψ as rank one operators.
(cid:13) (cid:13) 1 2
⊗ ⊗
A straightforward computation shows that ξ ε η if and only if T ε T . It follows from the
1 2
k k
elementary properties of rank one operators and Lemma 3.6 that
(ξ ψ)2 ξ 2 |(ξ|η)|2 if (ξ ψ)η = 0
M (ξ) = | | | k k − kηk2 | 6
T1,T2  (ξ ψ)2(cid:16)ξ 2 (cid:17) if (ξ ψ)η = 0.
| | | k k |
Thus we reach

ξ ε η T ε T
1 2
k ⇐⇒ k
sup M (ξ) : ξ C, ξ = 1 ε2 T 2
⇐⇒ { T1,T2 ∈ k k } ≤ k 1k
ξ 2 η 2 (ξ η)2 ε2 ξ 2 η 2
⇐⇒ k k k k −| | | ≤ k k k k
(ξ η) 1 ε2 ξ η
⇐⇒ | | | ≥ − k kk k
η 2ξ (pξ η)η ε ξ η 2.
⇐⇒ k k − | ≤ k kk k
(cid:13) (cid:13)
Further, by the equivalence (i) (cid:13)(iv) of Theorem(cid:13)3.7 yields the
⇔(cid:13) (cid:13)
ξ ε η T ε T
1 2
k ⇐⇒ k
sup (T ω ζ) : ω = ζ = 1, (T ω ζ)= 0 ε T
1 2 1
⇐⇒ {| | | k k k k | } ≤ k k
sup (ω ψ) (ξ ζ) : ω = ζ = 1, (ω ψ)(η ζ)= 0 ε ξ
⇐⇒ {| | || | | k k k k | | } ≤ k k
sup (ξ ζ) : ζ H , ζ = 1,(η ζ)= 0 ε ξ .
⇐⇒ {| | | ∈ k k | } ≤ k k
(cid:3)
Remark 3.9. If we choose ε = 0 in Corollary 3.8, we reach the fact that two vectors in a Hilbert
space are parallel if and only if they are proportional.
Next, we investigate the case when an operator is parallel to the identity operator.
Theorem 3.10. Let T B(H ). Then the following statements are equivalent:
∈
(i) T I;
k
(ii) T T∗;
k
(iii) There exist a sequence of unit vectors ξ in H and λ T such that
n
{ } ∈
lim Tξ λ T ξ = 0;
n n
n→∞ − k k
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)