Properties of quasi-similarity on θ-operators

Introduction

Let H denote a complex Hilbert space and B(H) denote a banach algebra of bounded linear operators on a Hilbert space H. An operator T ∈ B(H) is a structure preserving map or a symbol that represents a transformation that can be performed on a function. If P ∈ B(H), then P * denotes the adjoint of an operator P and P -1 denotes the inverse of P. If A, B ∈ B(H) then (AB) -1 = B -1 A -1 . Kiprop et al. (2021) [2] demonstrated that normal operators are θ-operators. Kamau et al. (2024) [6] noted that quasi-similarity preserves normality of an operator. Kiprop et al. (2021) [2] determined the properties of θ-operators on unitary equivalence and almost similarity of an operator. Kibe et al. (2024) [3] extended the work of Kiprop et al. (2021) [2] to the class of unitary quasi-equivalence and demonstrated that unitary quasi-equivalence relation preserves θ-operators.

Definition 1.1:

A Hilbert space H (Halmos, 2017) [5] . A Hilbert space H is a complete inner product space. Definition 1.2: Quasi-affinity operator (Nzimbi et al., 2016) [8] An operator X ∈ B (H, K) is said to be quasi-affinity or quasi-invertible if it is injective with dense range. Definition 1.3: Quasi-similar operators (Nagy., 2010) [7] . Two operators S and T of bounded linear operators B(H) and B(K) on Hilbert spaces H and K respectively are said to be quasisimilar if there exist quasi-affinity operators X ∈ B (H, K) and Y ∈ B(K, H) satisfying the equations XS = TX and SY = Y T .

Methodology

In order to achieve the desired results, the following outcomes were useful: Theorem 2.1: (Kiprop et al., 2021) [2] . If T is self adjoint, then

Example 2.2: Consider the operator 𝑇 = ( 7 1 1 9

) on the two-dimensional Hilbert space 𝐻 2 , we establish that the operator 𝑇 is a 𝜃operator.

Solution:

). We need to determine whether 𝑇 * 𝑇(𝑇 * + 𝑇) = (𝑇 * + 𝑇)𝑇 * 𝑇.

Lemma 2.4: (Kiprop et al., 2021) [2] . Suppose E, F ∈ B(H) such that E is unitary equivalent to F and E is a θ-operator, then F is also a θ-operator.

Lemma 2.5: (Kiprop et al., 2021) [2] . Suppose K, L ∈ B(H) such that K is almost similar to L and K is a θ-operator, then L is also a θ-operator.

Lemma 2.6: (Kibe et al., 2024) [3] . Suppose S, T ∈ B(H) such that S is unitary quasi-equivalent to T and S is a θ-operator, then T is also a θoperator.

Main Results Theorem 3.1

Suppose A, B ∈ B(H) such that A is quasi-similar to B and A is a θ-operator, then B is also a θ-operator.

Proof. Since A and B are quasi-similar, then there exists two operators X and Y that are quasi-affinity/ quasi-invertible such that

Considering equation ( 1), since X is quasi-invertible then pre-multiplying both sides by X -1 we have which follows that,

Since A is a θ-operator, then by definition

But from Theorem 2.3 since X is invertible, then (X -1 ) * = (X * ) -1 and equation 3 becomes

Given two operators A, B ∈ B(H), we have,

But X is quasi-invertible, hence XX * = I and equation ( 6) becomes

Since X is invertible, then XX -1 = X -1 X = I, XX * = I and from Theorem 2.3 (AB) * = B * A * , then equation ( 8) becomes

Pre-multiplying and post-multiplying both sides of equation ( 11) by X and X -1 respectively, we get

Applying equation (5) in equation ( 12), we get

But XX -1 = I and XX * = I, hence equation (13) becomes

Since B * B is common on both sides of equation (15), we have

which implies that B is also a θ-operator. Also, since Y is quasi-invertible then multiplying equation (2) from the right by Y -1 we have which follows that,

Since A is a θ-operator, then by definition

But from Theorem 2.3 since Y is invertible, then (Y -1 ) * = (Y * ) -1 and equation ( 16) becomes

Since Y is invertible, then Y Y -1 = Y -1 Y = I and from Theorem 2.3 and equation ( 5), then equation ( 18) becomes

) Pre-multiplying and post-multiplying both sides of equation ( 19) by Y -1 and Y respectively, we get which implies that B is also a θ-operator.

References

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