New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal

Definition 2.1

Let A ∈ ℬ ( ℋ ) . A is called normal if: A A ∗ = A ∗ A , hyponormal if: A A ∗ ≤ A ∗ A , p -hyponormal ( 0 < p ≤ 1 ) if: ( A A ∗ ) p ≤ ( A ∗ A ) p , paranormal if: ∣ ∣ A x ∣ ∣ 2 ≤ ∣ ∣ A 2 x ∣ ∣ ∣ ∣ x ∣ ∣ , for all x ∈ ℋ , normaloid if: r ( A ) = ∣ ∣ A ∣ ∣ (where r ( A ) denotes the spectral radius of A ) and log -hyponormal if: A is invertible and satisfies log ( A ∗ A ) ≥ log ( A A ∗ ) .

Definition 2.2

[22] Let λ ∈ C . We say that λ ∈ σ a r ( A ) (the approximate reduced spectrum of A ), if there is a normed sequence { x n } ∈ ℋ such that: lim n ( A − λ I ) x n = 0 and lim n ( A − λ I ) ∗ x n = 0 .

Definition 2.3

The numerical range of an operator A is defined by:

Definition 2.4

[18] An operator A ∈ ℬ ( ℋ ) is said to be ∗ -finite if:

Proposition 2.5

[18] These are equivalent conditions on an operator A .

1. A is ∗ -finite.

2. ∥ T A − A T ∗ − I ∥ ≥ 1 ; ∀ T ∈ ℬ ( ℋ ) .

Definition 3.1

Let A , B ∈ ℬ ( ℋ ) . The pair ( A , B ) is said to be a pair of generalized ∗ -finite operators if:

We note by Gℱ ∗ ( ℋ ) the class of generalized ∗ -finite operators, i.e.:

Theorem 3.2

Let A , B ∈ ℬ ( ℋ ) . These are equivalent conditions on A , B .

1. 0 ∈ W ( T A − B T ∗ ) ¯ ; ∀ T ∈ ℬ ( ℋ ) ;

2. ∥ T A − B T ∗ − λ I ∥ ≥ ∣ λ ∣ ; ∀ λ ∈ C , ∀ T ∈ ℬ ( ℋ ) .

Proof

In [11] the author has shown that for any operator A , we have: 0 ∈ W ( A ) ¯ iff: ∥ A − λ I ∥ ≥ ∣ λ ∣ , for all λ ∈ C . Replacing A by T A − B T ∗ we get:

Remark 3.3

( 0 , I ) is not a pair of generalized ∗ -finite operators. Otherwise 0 ∈ W ( − T ∗ ) ¯ , for all T ∈ ℬ ( ℋ ) . Moreover, if T = i I , we get: 0 ∈ W ( i I ) ¯ = { i } , which is a contradiction.

Remark 3.4

The condition (ii) above means that inf T ∥ T A − B T ∗ − λ I ∥ = ∣ λ ∣ , for all λ ∈ C .

Proposition 3.5

Let ( A , B ) ∈ Gℱ ∗ ( ℋ ) . Then ( α A , α B ) ∈ Gℱ ∗ ( ℋ ) , for all α ∈ C .

Proof

Clearly, the pair of null operators is a generalized pair of ∗ -finite operators. Let α ∈ C ∗ , given ε > 0 , then we have:

for all T ∈ ℬ ( ℋ ) and x ∈ ℋ . Thus,

i.e., 0 ∈ W ( T ( α A ) − ( α B ) T ∗ ) ¯ . □

Proposition 3.6

If ( B , A ) ∈ Gℱ ∗ ( ℋ ) , then ( A ∗ , B ∗ ) ∈ Gℱ ∗ ( ℋ ) .

Proof

Suppose that ( B , A ) ∈ Gℱ ∗ ( ℋ ) . Hence,

Since the map ℬ ( ℋ ) → ℬ ( ℋ ) : T → T ∗ is surjective, then:

So,

i.e., ( A ∗ , B ∗ ) is generalized ∗ -finite.□

Proposition 3.7

If ( A , B ),( C , D ) are pairs of generalized ∗ -finite operators, then ( A + C , B + D ) is generalized ∗ -finite.

Proof

Let ( A , B ) , ( C , D ) ∈ Gℱ ∗ ( ℋ ) . For ε > 0 , we have:

for all T ∈ ℬ ( ℋ ) and x ∈ ℋ . Hence,

Proposition 3.8

There exists ( A , B ) ∈ Gℱ ∗ ( ℋ ) , such that:

Proof

Suppose that ( A − λ I , B − λ I ) ∈ Gℱ ∗ ( ℋ ) . Then 0 ∈ W ( T ( A − λ I ) − ( B − λ I ) T ∗ ) ¯ for all T ∈ ℬ ( ℋ ) . Put ( A , B ) = ( 0 , 0 ) and T = i I , so 0 ∈ W ( − 2 λ i I ) ¯ = { − 2 λ i } , for all λ ∈ C ∗ , which is a contradiction.□

Theorem 3.9

Let ℋ = ℋ 1 ⊕ ℋ 2 and let A , B ∈ ℬ ( ℋ ) be operators of the form:

If for all i ; i = 1 , 2 ¯ : ( A i i , B i i ) ∈ Gℱ ∗ ( ℋ i ) , then ( A , B ) ∈ Gℱ ∗ ( ℋ ) .

Proof

We have for all X ∈ ℬ ( ℋ )

Then, for all λ ∈ C , we have:

Theorem 3.10

Gℱ ∗ ( ℋ ) is closed in ℬ ( ℋ ) in the operator norm topology.

Proof

Let ( A n , B n ) n ∈ N ∗ be a sequence in Gℱ ∗ ( ℋ ) such that A n → A and B n → B . Then,

for all T ∈ ℬ ( ℋ ) and all λ ∈ C . By letting n → + ∞ , we obtain ∥ T A − B T ∗ − λ I ∥ ≥ ∣ λ ∣ . Hence, ( A , B ) ∈ Gℱ ∗ ( ℋ ) .□

Proposition 3.11

Gℱ ∗ ( ℋ ) is not dense in ℬ ( ℋ ) in the operator norm topology.

Proof

Theorem 3.10 and Remark 3.3 show that Gℱ ∗ ( ℋ ) cannot be dense in ℬ ( ℋ ) in the operator norm topology since Gℱ ∗ ( ℋ ) ¯ ≠ ℬ ( ℋ ) .□

Proposition 3.12

Gℱ ∗ ( ℋ ) is invariant under unitary equivalence.

Proof

Let ( A , B ) ∈ Gℱ ∗ ( ℋ ) and U is a unitary operator. Then by Theorem 3.2 we have:

Lemma 3.13

Let A , B be finite rank operators, then ( A , B ) is generalized ∗ -finite.

Proof

For a finite rank operator T ∈ ℬ ( ℋ ) , T A − B T ∗ is also of finite rank. Since 0 belongs to the spectrum of each finite rank operator, then 0 ∈ σ ( T A − B T ∗ ) ⊆ W ( T A − B T ∗ ) ¯ , for all T ∈ ℬ ( ℋ ) . Consequently, ( A , B ) is generalized ∗ -finite.□

Theorem 3.14

Let A , B ∈ ℬ ( ℋ ) . If there exist a normed sequence ( x n ) n ≥ 1 ⊂ ℋ and some scalar λ verifying:

then ( A − λ I , B − λ I ) is generalized ∗ -finite.

Proof

We have for all T ∈ ℬ ( ℋ ) :

Letting n → ∞ , we get ∥ T ( A − λ I ) − ( B − λ I ) T ∗ − λ I ∥ ≥ ∣ λ ∣ , for all T ∈ ℬ ( ℋ ) and all λ ∈ C .□

Corollary 3.15

If A ∈ ℬ ( ℋ ) , then for all λ ∈ σ a ( A ) and for all C ∈ ℬ ( ℋ ) ,

Proof

Let λ ∈ σ a ( A ) . Then there exists a normed sequence ( x n ) n ≥ 1 in ℋ verifying: ∥ ( A − λ I ) x n ∥ → 0 . If T = A − λ I and S = C T with C ∈ ℬ ( ℋ ) , then:

and

Thus,

Lemma 3.16

[23] If A is paranormal, then σ a r ( A ) ≠ ϕ .

Proposition 3.17

If A , B ∈ ℬ ( ℋ ) are paranormal operators, then ( A − λ I , B − λ I ) ∈ Gℱ ∗ ( ℋ ) , for all λ ∈ σ a r ( A ) ∩ σ a r ( B ) .

Proof

Let A , B ∈ ℬ ( ℋ ) be paranormal operators. Then by Lemma 3.16 there exist λ ∈ σ a r ( A ) ∩ σ a r ( B ) and a normed sequence ( x n ) n ≥ 1 in ℋ satisfying:

Hence, by Theorem 3.14 we get ( A − λ I , B − λ I ) ∈ Gℱ ∗ ( ℋ ) .□

Corollary 3.18

In all the next cases, the pair ( A − λ I , B − λ I ) is generalized ∗ -finite for all λ ∈ σ a r ( A ) ∩ σ a r ( B ) .

1. A and B are hyponormal operators,

2. A and B are p-hyponormal operators,

3. A and B are class A operators,

4. A and B are log-hyponormal operators.

Proposition 3.19

Let A , B ∈ ℬ ( ℋ ) . If R A , B contains no invertible operator, then ( A , B ) ∈ Gℱ ∗ ( ℋ ) .

Proof

Suppose that C ∈ R A , B (i.e., C = T A − B T ∗ ) is a noninvertible operator. Thus by [24, p. 162]: ∥ C − λ I ∥ ≥ ∣ λ ∣ , ∀ λ ∈ C . Then ( A , B ) ∈ Gℱ ∗ ( ℋ ) .□

Proposition 3.20

If ( A , B ) ∈ Gℱ ∗ ( ℋ ) , then R A , B ¯ (the closure of R A , B ) contains no nonzero scalar operator.

Proof

Suppose that λ I ∈ R A , B ¯ for some λ ∈ C ∗ , then there exists a sequence ( T n ) in ℬ ( ℋ ) satisfying

Since ( A , B ) ∈ Gℱ ∗ ( ℋ ) , ∣ λ ∣ ≤ ∥ T n A − B T n ∗ − λ I ∥ → 0 , a contradiction.□