A note on -Isometries and self-similarity of operators on Hilbert spaces

In this paper we investigate -isometries and related classes of operators. We will also introduce and study the notions of similitudes, self-similarity and -metric equivalence relations of operators. It will be shown that self-similarity implies -metric equivalence of operators. We will characterize -isometric and -unitary and prove that quasisimilar -isometries are unitarily equivalent.

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Introduction

Let 𝐻 denote a Hilbert space and 𝐵(𝐻) denote the Banach algebra of bounded linear operators. If 𝑇 ∈ 𝐵(𝐻), then 𝑇 * denotes the adjoint of 𝑇 while 𝐾𝑒𝑟(𝑇), 𝑅𝑎𝑛(𝑇), 𝑀and 𝑀 ⊥ stands for the kernel of𝑇, range of , closure of 𝑀 and orthogonal complement of a closed subspace 𝑀 of𝐻, respectively. We denote by 𝜎(𝑇), ‖𝑇‖, 𝑟(𝑇) and 𝑊(𝑇) the spectrum, norm, spectral radius of and numerical range of 𝑇, respectively. Recall that an operator 𝑇 ∈ 𝐵(𝐻) is normal if 𝑇 * 𝑇 = 𝑇𝑇 * , unitary if 𝑇 * 𝑇 = 𝑇𝑇 * = 𝐼, a projection(or idempotent) if 𝑇 2 = 𝑇, an orthogonal projection if 𝑇 2 = 𝑇and 𝑇 * = 𝑇, an isometry if 𝑇 * 𝑇 = 𝐼 (equivalently‖𝑇𝑥 -𝑇𝑦‖ = ‖𝑥 -𝑦‖, for all 𝑥, 𝑦 ∈ 𝐻), a partial isometry if 𝑇𝑇 * 𝑇 = 𝑇, quasinormal if (𝑇 * 𝑇)𝑇 = 𝑇(𝑇 * 𝑇), a scalar if 𝑇 = 𝛼𝐼 where 𝛼 ∈ ℂ, a contraction if ‖𝑇𝑥‖ ≤ ‖𝑥‖, for all𝑥 ∈ 𝐻. Two operators 𝐴 ∈ 𝐵(𝐻) and 𝐵 ∈ 𝐵(𝐾) are said to be similar if there exists an invertible operator 𝑁 ∈ 𝐵(𝐻, 𝐾) such that 𝑁𝐴 = 𝐵𝑁or equivalently𝐴 = 𝑁 -1 𝐵𝑁, and are unitarily equivalent if there exists a unitary operator 𝑈 ∈ 𝐵 + (𝐻, 𝐾) (Banach algebra of all invertible operators in𝐵(𝐻)) such that 𝑈𝐴 = 𝐵𝑈 (i.e.𝐴 = 𝑈 * 𝐵𝑈equivalently, 𝐴 = 𝑈 -1 𝐵𝑈). An operator 𝑋 ∈ 𝐵(𝐻, 𝐾) is a quasiaffinity if it is injective with dense range. Two operators 𝐴 ∈ 𝐵(𝐻) and 𝐵 ∈ 𝐵(𝐾) are quasisimilar if there exist quasiaffinities 𝑋 ∈ 𝐵(𝐻, 𝐾) and 𝑌 ∈ 𝐵(𝐾, 𝐻) such that 𝑋𝐴 = 𝐵𝑋 and 𝐴𝑌 = 𝑌𝐵. Two operators 𝐴, 𝐵 ∈ 𝐵(𝐻) are said to be metrically equivalent if‖𝐴𝑥‖ = ‖𝐵𝑥‖, (equivalently, |⟨𝐴𝑥, 𝐴𝑥⟩| 1 2 = |⟨𝐵𝑥, 𝐵𝑥⟩| 1 2 for all𝑥 ∈ 𝐻, which is equivalent to having 𝐴 * 𝐴 = 𝐵 * 𝐵). The notion of metric equivalence of operators was introduced by Nzimbi et al. [6] . Clearly similarity, unitary equivalence and metric equivalence are equivalence relations on𝐵(𝐻). A similitude is an operator 𝑇 ∈ 𝐵(𝐻) that is a positive scalar multiple of an isometry (equivalently, if ‖𝑇𝑥 -𝑇𝑦‖ = 𝛼‖𝑥 -𝑦‖, for all 𝑥, 𝑦 ∈ 𝐻 and 𝛼 > 0). Clearly every isometry is a similitude. Similitudes are obtained as compositions of rotation, translation and scaling (dilation) operators. Such operators are similarities in the geometric sense, in that they map sets to similar sets and so preserve shapes up to a scaling factor. Similitudes are extensions of isometries. The theory of isometries has been extensively studied (see [2] , [4] and [5] for more exposition).

Main results Theorem 2.1

The product of similitudes is a similitude.

T T

Proof. Suppose 𝐴, 𝐵 ∈ 𝐵(𝐻) such that 𝐴 = 𝛼𝑉 and𝐵 = 𝛽𝑊, where 𝛼, 𝛽 > 0 and 𝑉, 𝑊are isometries. Then 𝐴𝐵 = 𝛼𝛽𝑉𝑊. The rest of the proof follows from the fact that the product of isometries is an isometry. Theorem 2.1 says that the set of similitudes is a semigroup. Theorem 2.2 If 𝐴, 𝐵 ∈ 𝐵(𝐻) are quasisimilar similitude operators, then they are similar. Proof. Suppose, without loss of generality, that 𝐴 = 𝛼𝑈and 𝐵 = 𝛼𝑉, where 𝑈, 𝑉 are isometries and 𝛼 > 0. Using the fact that every isometry is quasinormal and ( [9] , Proposition 4.2), we conclude that any quasisimilar isometries in a Hilbert space are unitarily equivalent. Suppose𝑈 = 𝑊𝑉𝑊 * , where 𝑊is unitary. Then𝐴 = 𝛼𝑈 = 𝛼𝑊( 1 𝛼 𝐵)𝑊 * = 𝑊𝐵𝑊 * = 𝑊𝐵𝑊 -1 . This proves the claim. A consequence of Theorem 2.2 is that quasisimilar similitudes are unitarily equivalent. Definition 2.3 Two operators 𝐴, 𝐵 ∈ 𝐵(𝐻) are said to be self-similar denoted by 𝐴 ∼ 𝑠.𝑠 𝐵 if there is a similitude 𝑆 ∈ 𝐵(𝐻) such that𝐵 = 𝑆𝐴. A closed subspace ⊆ 𝐻is said to be self-similar if it is invariant under a similitude𝑇. Remark. We note that the trivial subspaces 𝑀 1 = {0}and 𝑀 2 = 𝐻are invariant under any bounded linear operator on 𝐻and hence are self-similar.

In particular, if𝜀 = 0, then the class of 0 -isometries contains the class of isometries and contraction operators. A similitude need not be invertible, but if it is invertible, then it is a multiple if a unitary operator. Theorem 2.8 [5] An operator 𝑇 is a multiple of an isometry if and only if 𝑇 * 𝑇 is a scalar. Theorem 2.9 If 𝑇 is a similitude, then 𝜎(|𝑇|) = {𝛼} for some 𝛼 > 0. Proof. This follows from the fact that 𝑇 is a similitude if and only if |𝑇| = 𝛼𝐼, for some 𝛼 > 0. Definition 2.10 An operator 𝑇 ∈ 𝐵(𝐻) is called a 𝛼 -projection if 𝑇 = 𝛼𝑃, for some projection 𝑃 ∈ 𝐵(𝐻). An operator 𝑇 ∈ 𝐵(𝐻) is called a 𝛼 -isometry if 𝑇 = 𝛼𝑉, where 𝑉is an isometry and 𝛼 ∈ ℂ. It is clear that if 𝐴 ∈ 𝐵(𝐻) is an 𝛼 -isometry and 𝑈is an isometry, then 𝐵 = 𝑈𝐴𝑈 * is a 𝛼 -projection. It is also clear that an 𝛼 -projection is a projection if 𝛼 ∈ {0,1}. It is also clear that a similitude is an 𝛼 -isometry. In fact the following inclusion holds: 𝑆𝑖𝑚𝑖𝑙𝑖𝑡𝑢𝑑𝑒 ⊂ 𝜀 -𝑖𝑠𝑜𝑚𝑒𝑡𝑟𝑦 ⊂ 𝛼 -𝑖𝑠𝑜𝑚𝑒𝑡𝑟𝑦.

Theorem 2.11

If 𝐴 ∈ 𝐵(𝐻) is unitarily equivalent to a similitude, then it is a similitude. Proof. Suppose that 𝐴 = 𝑈𝐵𝑈 * , where 𝐵 = 𝛼𝑉 with 𝑉 isometric. Then a simple computation shows that 𝐴 = 𝑈𝐵𝑈 * = 𝑈(𝛼𝑉)𝑈 * = 𝛼𝑈𝑉𝑈 * . Now, if we let 𝑊 = 𝑈𝑉𝑈 * , it is clear that 𝑊is an isometry. Therefore 𝐴 = 𝛼𝑊. This establishes the claim.

Example 1 For 𝛼 > 0, the operator 𝑇acting on ℓ 2 (ℕ) defined by 𝑇(𝑥 1 , 𝑥 2 , 𝑥 3 , ⋯ ) = (0, 𝛼𝑥 1 , 𝛼𝑥 2 , 𝛼𝑥 3 , ⋯ ) is a similitude. Clearly, 𝑇 = 𝛼𝑉, where 𝑉is the unilateral shift on ℓ 2 (ℕ). In fact the set of similitudes is a multiplicative group.

Theorem 2.14 Self-similarity is an equivalence relation in the Banach subalgebra of invertible operators. Proof. Recall that two operators 𝐴, 𝐵 ∈ 𝐵(𝐻) are self-similar if there exists a similitude 𝑆 ∈ 𝐵(𝐻) such that 𝐴 = 𝑆𝐵. Clearly, 𝐴 = 𝐼𝐴. Thus 𝐴 ∼ 𝑠.𝑠 𝐴. Suppose 𝐴, 𝐵 ∈ 𝐵(𝐻) are invertible and 𝐴 ∼ 𝑠.𝑠 𝐵. Then there exists an invertible similitude𝑆 ∈ 𝐵(𝐻)

such that 𝐴 = 𝑆𝐵. Using Proposition 2.13, we have that 𝐵 = 𝑆 -1 𝐴. This shows that 𝐵 ∼

𝐾𝑒𝑟(𝐵) ≠ 𝐾𝑒𝑟(𝐴).

A simple computation shows that these operators are similar but not self-similar. This shows that there exist similar operators which are not self-similar. We also note that while similarity preserves invertibility, self-similarity need not preserve invertibility. To see this, let 𝐴 = 𝐼and 𝐵 = 𝑆, the unilateral shift on ℓ 2 (ℕ). Clearly,𝐴 and 𝐵are self-similar but are not both invertible.

Theorem 3.3 If 𝐴, 𝐵 ∈ 𝐵(𝐻) are similar and self-similar, then 𝐴 = 𝐵.

Theorem 3.4 Self-similar operators are 𝛼 -metrically equivalent.

Proof. Suppose that 𝐴, 𝐵 ∈ 𝐵(𝐻) are such that 𝐴 = 𝑆𝐵for some similitude𝑆 = 𝛼𝑉, where 𝑉is an isometry and 𝛼 > 0. A simple computation gives 𝐴 * 𝐴 = 𝛼 2 𝐵 * 𝐵, which establishes the claim. We note that if 𝑇 ∈ 𝐵(𝐻, 𝐾) is an 𝛼 -isometry, then ⟨𝑇𝑥, 𝑇𝑦⟩ = ⟨𝛼𝑥, 𝛼𝑦⟩ = |𝛼| 2 ⟨𝑥, 𝑦⟩, for all 𝑥 ∈ 𝐻and𝑦 ∈ 𝐾. In this case, we say that 𝑇is 𝛼 -isometric. Clearly, every unitary operator is an 𝛼 -unitary and every 𝛼 -unitary is an 𝛼 -isometry. That is

The converse is not true in general. The unilateral shift on ℓ 2 (ℕ) is an 𝛼 -isometry which is not an 𝛼 -unitary. Example 4. Suppose that in the Hilbert space 𝐻 = ℝ 2 , two closed and connected subspaces 𝛺 1 and 𝛺 2 are similar, with a scaling factor 𝛼 > 0. Then there exists a similitude𝑇: 𝛺 1 → 𝛺 2 . A simple computation shows that the perimeter ratio is 𝛼and the area ratio is 𝛼 2 .

Remark. Example 4 can be extended to𝛼 -isometries (that is, all𝛼 ∈ ℂ). In any case, the distance between points is |𝛼|. This is the ratio of magnification or dilation factor or similitude factor.

Discussion

Similitudes that are contractions (that is, 𝜶 -isometries where the scaling factor0 < 𝛼 < 1 ) are applicable in the study of fractals and are a popular tool to provide existence of solutions in many problems. Self-similarity is a characterization of fractals arising from iterated function systems and find application in chaotic dynamical systems and modelling of systems (see [10] ). Similitudes in general reduce or increase distance between points and are very applicable in the study and application of frames, specifically generalized or tight 𝑔 -frames ( [1] , Definition 3.1) and wavelets in image compression and decompression schemes, geometry and DNA sequences, gene mutation and fractals (see [8] ).

Conclusion

In this paper, the notions of 𝛼 -isometry and self-similarity are introduced. It has been shown that self-similarity is an equivalence relation. It has also been shown that quasisimilar 𝛼 -isometries are unitarily equivalent. It is also shown that the commutant of an 𝛼unitary operator T consists of isometries.

Acknowledgements

The author wishes to pay tribute to the Eastern Africa University Mathematics Programme (EAUMP) for funding this research.

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