Characteristic functions of some contraction operators

Introduction

Let ℋ denote a Hilbert space and 𝐵𝐵(ℋ) denote the Banach algebra of bounded linear operators. If 𝑇𝑇 ∈ 𝐵𝐵(ℋ), then 𝑇𝑇 * denotes the adjoint of 𝑇𝑇, while 𝐾𝐾𝐾𝐾𝐾𝐾(𝑇𝑇), 𝑅𝑅𝑅𝑅𝑅𝑅(𝑇𝑇), ℳ and ℳ ⊥ stands for the kernel of 𝑇𝑇, range of 𝑇𝑇, closure of ℳ and orthogonal complement of a closed subspace ℳ of ℋ, respectively. We denote by 𝜎𝜎(𝑇𝑇), ∥ 𝑇𝑇 ∥ and 𝑊𝑊(𝑇𝑇), the spectrum, norm and numerical range of 𝑇𝑇, respectively. By 𝔻𝔻 = {𝜆𝜆 ∈ ℂ: |𝜆𝜆| < 1} and 𝜕𝜕𝔻𝔻 = {𝜆𝜆 ∈ ℂ: |𝜆𝜆| = 1}, we denote the open unit disk and the unit circle on the complex plane, respectively. An operator 𝑇𝑇 ∈ 𝐵𝐵(ℋ) is unitary if 𝑇𝑇 * 𝑇𝑇 = 𝑇𝑇𝑇𝑇 * = 𝐼𝐼. an isometry if 𝑇𝑇 * 𝑇𝑇 = 𝐼𝐼. a co-isometry if 𝑇𝑇𝑇𝑇 * = 𝐼𝐼. a contraction if ∥ 𝑇𝑇 ∥≤ 1. nilpotent if 𝑇𝑇 𝑛𝑛 = 0 for some positive integer 𝑅𝑅. The least such 𝑅𝑅 is called the order of nilpotence of 𝑇𝑇.

Two operators 𝐴𝐴 ∈ 𝐵𝐵(ℋ) and 𝐵𝐵 ∈ 𝐵𝐵(𝒦𝒦) are said to be similar (denoted 𝐴𝐴 ∼ 𝐵𝐵) if there exists an invertible operator 𝑁𝑁 ∈ 𝐵𝐵(ℋ, 𝒦𝒦) such that 𝑁𝑁𝐴𝐴 = 𝐵𝐵𝑁𝑁 or equivalently 𝐴𝐴 = 𝑁𝑁 -1 𝐵𝐵𝑁𝑁, and are unitarily equivalent (denoted by 𝐴𝐴 ≅ 𝐵𝐵)if there exists a unitary operator 𝑈𝑈 ∈ 𝐵𝐵 + (ℋ, 𝒦𝒦) (Banach algebra of all invertible operators in 𝐵𝐵(ℋ)) such that 𝑈𝑈𝐴𝐴 = 𝐵𝐵𝑈𝑈 (i.e. 𝐴𝐴 = 𝑈𝑈 * 𝐵𝐵𝑈𝑈, equivalently, 𝐴𝐴 = 𝑈𝑈 -1 𝐵𝐵𝑈𝑈). Two operators 𝐴𝐴 ∈ 𝐵𝐵(ℋ) and 𝐵𝐵 ∈ 𝐵𝐵(𝒦𝒦) are said to be metrically equivalent (denoted by 𝐴𝐴 ∼ 𝑚𝑚 𝐵𝐵) if ∥ 𝐴𝐴𝐴𝐴 ∥=∥ 𝐵𝐵𝐴𝐴 ∥, (equivalently, |〈𝐴𝐴𝐴𝐴, 𝐴𝐴𝐴𝐴〉| for all 𝐴𝐴 ∈ ℋ). The concept of metric equivalence of operators was introduced in [7] . Clearly similarity, unitary equivalence and metric equivalence are equivalence relations on 𝐵𝐵(ℋ). Given a contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ), both (𝐼𝐼 -𝑇𝑇 * 𝑇𝑇) and (𝐼𝐼 -𝑇𝑇𝑇𝑇 * ) are positive operators and hence have unique square roots. We define 𝐷𝐷 𝑇𝑇 = (𝐼𝐼 -𝑇𝑇 * 𝑇𝑇) 1 2 and 𝐷𝐷 𝑇𝑇 * = (𝐼𝐼 -𝑇𝑇𝑇𝑇 * ) 1 2 and call them the defect operators of 𝑇𝑇. Clearly, 0 ≤ 𝐷𝐷 𝑇𝑇 , 𝐷𝐷 𝑇𝑇 * ≤ 𝐼𝐼. The closures of their ranges 𝒟𝒟 𝑇𝑇 = 𝐷𝐷 𝑇𝑇 (ℋ) and 𝒟𝒟 𝑇𝑇 * = 𝐷𝐷 𝑇𝑇 * (ℋ) are called the defect spaces of 𝑇𝑇. The respective dimensions(ranks) 𝑑𝑑 𝑇𝑇 and 𝑑𝑑 𝑇𝑇 * are called the defect indices of 𝑇𝑇. A contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ) is said to be completely non-unitary(c.n.u) if there exists no nontrivial reducing subspace of ℳ ⊂ ℋ of 𝑇𝑇 on which 𝑇𝑇 acts unitarily (i.e. 𝑇𝑇| ℳ is unitary), or equivalently if its unitary part acts on the zero space {0}. Clearly, a contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ) is c.n.u if 𝑇𝑇 restricted to every (non-zero 𝑇𝑇-reducing) subspace of ℋ is non-unitary.

It was observed in [5] that 𝑑𝑑 𝑇𝑇 = 0 characterizes isometric operators, whereas 𝑑𝑑 𝑇𝑇 = 𝑑𝑑 𝑇𝑇 * = 0 characterizes unitary operators and that these numbers measure, in a sense, the deviation of the contraction 𝑇𝑇 from being unitary. For any contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ) there exists a unique decomposition ℋ = ℳ 0 ⊕ ℳ 1 such that 𝑇𝑇ℳ 0 ⊆ ℳ 0 and 𝑇𝑇ℳ 1 ⊆ ℳ 1 and 𝑇𝑇 0 = 𝑇𝑇| ℳ 0 is unitary, whereas 𝑇𝑇 1 = 𝑇𝑇| ℳ 1 is completely non-unitary (c.n.u). Thus 𝑇𝑇 has a unique decomposition as 𝑇𝑇 = 𝑇𝑇 0 ⊕ 𝑇𝑇 1 . This is the Nagy-Foia𝑠𝑠-Langer Decomposition (see) [4] . With this decomposition, we observe that 𝐷𝐷 𝑇𝑇 = 0 ⊕ 𝐷𝐷 𝑇𝑇 1 = 𝐷𝐷 𝑇𝑇 1 and 𝐷𝐷 𝑇𝑇 * = 0 ⊕ 𝐷𝐷 𝑇𝑇 1 * = 𝐷𝐷 𝑇𝑇 1 * (where the second equality denotes "identified with") and 𝒟𝒟 𝑇𝑇 = 𝒟𝒟 𝑇𝑇 1 and 𝒟𝒟 𝑇𝑇 * = 𝒟𝒟 𝑇𝑇 1 * . Obviously 𝑇𝑇 is unitary if and only if 𝒟𝒟 𝑇𝑇 = 𝒟𝒟 𝑇𝑇 0 = {0}.

Considering the asymptotic behaviour of contractions, Sz.-Nagy and Foias [5] introduced the following classes: A contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ) is of class: 𝐶𝐶 1. if ∥ 𝑇𝑇 𝑛𝑛 𝐴𝐴 ∥→ 0, strongly as 𝑅𝑅 → ∞ for every 0 ≠ 𝐴𝐴 ∈ ℋ. 𝐶𝐶 .1 if ∥ 𝑇𝑇 * 𝑛𝑛 𝐴𝐴 ∥→ 0, strongly as 𝑅𝑅 → ∞ for every 0 ≠ 𝐴𝐴 ∈ ℋ. 𝐶𝐶 0. if ∥ 𝑇𝑇 𝑛𝑛 𝐴𝐴 ∥→ 0, strongly as 𝑅𝑅 → ∞ for every 𝐴𝐴 ∈ ℋ. 𝐶𝐶 .0 if ∥ 𝑇𝑇 * 𝑛𝑛 𝐴𝐴 ∥→ 0, strongly as 𝑅𝑅 → ∞ for every 𝐴𝐴 ∈ ℋ. 𝐶𝐶 𝑖𝑖𝑖𝑖 if 𝑇𝑇 ∈ 𝐶𝐶 𝑖𝑖. ∩ 𝐶𝐶 .𝑖𝑖 , 𝑖𝑖, 𝑗𝑗 = 0,1.

An operator 𝑇𝑇 ∈ 𝐵𝐵(ℋ) is called a proper (or pure, strict) contraction if ∥ 𝑇𝑇𝐴𝐴 ∥<∥ 𝐴𝐴 ∥, for every 𝐴𝐴 ∈ ℋ. An operator-valued analytic function Θ: 𝔻𝔻 ⟶ 𝐵𝐵(ℋ, 𝒦𝒦) is said to be contractive if ∥ Θ(𝜆𝜆)𝐴𝐴 ∥≤∥ 𝐴𝐴 ∥, for all 𝐴𝐴 ∈ ℋ and 𝜆𝜆 ∈ 𝔻𝔻 and purely contractive if ∥ Θ(0)𝑦𝑦 ∥<∥ 𝑦𝑦 ∥, for all 0 ≠ 𝑦𝑦 ∈ ℋ.

Main Results

To every completely non-unitary (c.n.u) contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ), Sz.-Nagy and Foia𝑠𝑠 [5] associated a contraction-valued holomorphic function Θ 𝑇𝑇 on the open unit disk 𝔻𝔻 = {𝜆𝜆 ∈ ℂ: |𝜆𝜆| < 1} on the complex plane. Such a function Θ 𝑇𝑇 (𝜆𝜆): 𝒟𝒟 𝑇𝑇 ⟶ 𝒟𝒟 𝑇𝑇 * defined for all 𝜆𝜆 ∈ 𝔻𝔻 is given by

This means that the values of Θ 𝑇𝑇 (𝜆𝜆) can be regarded as bounded operators from 𝒟𝒟 𝑇𝑇 into 𝒟𝒟 𝑇𝑇 * . We note that the characteristic function of a c.n.u. contraction 𝑇𝑇 is equivalent to the function in ℍ ∞ (𝐵𝐵(𝒟𝒟 𝑇𝑇 , 𝒟𝒟 𝑇𝑇 * ) given by the same formula.

For any c.n.u. contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ), it is clear that 𝑇𝑇𝐷𝐷 𝑇𝑇 = 𝐷𝐷 𝑇𝑇 * 𝑇𝑇.

Remark. The Neumann series for (𝐼𝐼

𝑛𝑛=1 𝜆𝜆 𝑛𝑛 𝑇𝑇 𝑛𝑛 for every 𝜆𝜆 ∈ ℂ. Since 𝜆𝜆 ∈ 𝔻𝔻, we have |𝜆𝜆| < 1. Clearly since 𝑇𝑇 is a contraction ∥ 𝜆𝜆𝑇𝑇 ∥= |𝜆𝜆| < 1, the Neumann series converges.

The characteristic function of a c.n.u contraction 𝑇𝑇 is defined for the values 𝜆𝜆 for which the operator (𝐼𝐼 -𝜆𝜆𝑇𝑇 * ) is boundedly invertible. We say that two operators 𝐴𝐴 ∈ 𝐵𝐵(ℋ 1 , ℋ 2 ) and 𝐵𝐵 ∈ 𝐵𝐵(𝒦𝒦 1 , 𝒦𝒦 2 ) coincide if there exist unitary operators 𝑈𝑈: ℋ 2 ⟶ ℋ 1 and 𝑉𝑉: 𝒦𝒦 1 ⟶ 𝒦𝒦 2 such that 𝑉𝑉𝐴𝐴𝑈𝑈 = 𝐵𝐵. Operator coincidence is an equivalence relation which is weaker than unitary equivalence. The operator-valued functions Θ(𝜆𝜆): 𝔻𝔻 ⟶ 𝐵𝐵(ℋ 1 , ℋ 2 ) and Ψ(𝜆𝜆): 𝔻𝔻 ⟶ 𝐵𝐵(𝒦𝒦 1 , 𝒦𝒦 2 ) are said to coincide (denoted by Θ ≅ Ψ) if there exist unitary operators 𝑈𝑈: ℋ 1 ⟶ 𝒦𝒦 1 and 𝑉𝑉: ℋ 2 ⟶ 𝒦𝒦 2 such that 𝑉𝑉Θ(𝜆𝜆) = Ψ(𝜆𝜆)𝑈𝑈, for all 𝜆𝜆 ∈ 𝔻𝔻. Moreover, for a given 𝐵𝐵(ℋ 1 , ℋ 2 )-valued purely contractive analytic function Θ(𝜆𝜆) defined on 𝔻𝔻, there exists a c.n.u. contraction 𝑇𝑇 on some Hilbert space such that Θ 𝑇𝑇 (𝜆𝜆) coincides with Θ(𝜆𝜆).

The mapping

is a contraction-valued (i.e. sup 𝜆𝜆∈𝔻𝔻 ∥ Θ 𝑇𝑇 (𝜆𝜆) ∥≤ 1), analytic function and by Wu [8] , Θ 𝑇𝑇 (0) = -𝑇𝑇| 𝒟𝒟 𝑇𝑇 is a pure contraction. With the decomposition of a contraction 𝑇𝑇 ∈ 𝐵𝐵(ℋ) given above Θ 𝑇𝑇 decomposes as Θ 𝑇𝑇 (𝜆𝜆) = Θ 𝑢𝑢 (𝜆𝜆) ⊕ Θ 0 (𝜆𝜆), where Θ 𝑢𝑢 is a unitary constant (i.e. if Θ(0) is a unitary operator from ℋ onto ℋ * , then Θ(𝜆𝜆) = Θ(0) for all 𝜆𝜆 ∈ 𝔻𝔻, function Θ is called a unitary constant) and Θ 0 is pure (and hence called the pure part of Θ), i.e. ∥ Θ 0 (0)𝐴𝐴 ∥<∥ 𝐴𝐴 ∥ for all 0 ≠ 𝐴𝐴 ∈ ℋ. Thus, for all 𝜆𝜆 ∈ 𝔻𝔻, Θ(𝜆𝜆)ℳ 0 ⊆ ℳ 0 and Θ(𝜆𝜆)ℳ 1 ⊆ ℳ 1 . For every contractive function Θ(𝜆𝜆): 𝔻𝔻 ⟶ 𝐵𝐵(𝒟𝒟 𝑇𝑇 , 𝒟𝒟 𝑇𝑇 * ), there exists a unique decomposition 𝒟𝒟 𝑇𝑇 = 𝒟𝒟 𝑇𝑇 0 ⊕ 𝒟𝒟 𝑇𝑇 1 , 𝒟𝒟 𝑇𝑇 * = 𝒟𝒟 𝑇𝑇 0 * ⊕ 𝒟𝒟 𝑇𝑇 1 * , so that for every fixed 𝜆𝜆, Θ 0 (𝜆𝜆): = Θ(𝜆𝜆)| 𝒟𝒟 𝑇𝑇 0 has range in 𝒟𝒟 𝑇𝑇 0 * and that Θ 1 (𝜆𝜆): = Θ(𝜆𝜆)| 𝒟𝒟 𝑇𝑇 1 has range in 𝒟𝒟 𝑇𝑇 1 * and that Θ 0 (𝜆𝜆): 𝔻𝔻 ⟶ 𝐵𝐵(𝒟𝒟 𝑇𝑇 0 , 𝒟𝒟 𝑇𝑇 0 * ) is a unitary constant and that Θ 1 (𝜆𝜆): 𝔻𝔻 ⟶ 𝐵𝐵(𝒟𝒟 𝑇𝑇 1 , 𝒟𝒟 𝑇𝑇 1 * ) is a purely contractive function.

Theorem 2.1 [5] Let 𝑇𝑇 ∈ 𝐵𝐵(ℋ) be a c.n.u contraction. Then ∥ 𝛩𝛩 𝑇𝑇 (0)𝐴𝐴 ∥<∥ 𝐴𝐴 ∥ for all 0 ≠ 𝐴𝐴 ∈ 𝒟𝒟 𝑇𝑇 . Theorem 2.2 [5] Two contractions 𝑇𝑇 1 ∈ 𝐵𝐵(ℋ 1 ) and 𝑇𝑇 2 ∈ 𝐵𝐵(ℋ 2 ) are unitarily equivalent if their characteristic functions 𝛩𝛩 𝑇𝑇 1 and 𝛩𝛩 𝑇𝑇 2 coincide.

Proof. Suppose 𝑇𝑇 2 = 𝑈𝑈𝑇𝑇 1 𝑈𝑈 * , for some unitary operator 𝑈𝑈 ∈ 𝐵𝐵(ℋ 1 , ℋ 2 ). By definition, 𝐷𝐷 𝑇𝑇 2 = 𝑉𝑉𝐷𝐷 𝑇𝑇 1 and 𝐷𝐷 𝑇𝑇 2 * = 𝑊𝑊𝐷𝐷 𝑇𝑇 1 * and Θ 𝑇𝑇 2 (𝜆𝜆) = 𝑊𝑊Θ 𝑇𝑇 1 (𝜆𝜆)𝑉𝑉 * , where 𝑉𝑉 = 𝑈𝑈| 𝐷𝐷 𝑇𝑇 1 , and 𝑊𝑊 = 𝑈𝑈| 𝐷𝐷 𝑇𝑇 1 * . Clearly 𝑉𝑉 and 𝑊𝑊 are unitary. This establishes the claim.

The converse of Theorem 2.2 is not true in general. The characteristic function of a c.n.u contraction 𝑇𝑇 is said to be constant if Θ 𝑇𝑇 (𝜆𝜆) = Θ 𝑇𝑇 (0), for every 𝜆𝜆 ∈ 𝔻𝔻.

Theorem 2.3 If 𝑇𝑇 ∈ 𝐵𝐵(ℋ) is an isometry then the characteristic function 𝛩𝛩 𝑇𝑇 is identically zero almost everywhere.

Proof. We first prove the result for a unitary 𝑇𝑇. For a unitary operator 𝑇𝑇, 𝐷𝐷 𝑇𝑇 = 𝐷𝐷 𝑇𝑇 * = 0 and 𝒟𝒟 𝑇𝑇 = 𝐷𝐷 𝑇𝑇 ℋ = {0}. Therefore, Θ 𝑇𝑇 = -𝑇𝑇| 𝒟𝒟 𝑇𝑇 = -𝑇𝑇| {0} = 0, for all 𝜆𝜆 ∈ 𝔻𝔻. Since 𝐷𝐷 𝑇𝑇 = 0 for a unilateral shift 𝑇𝑇, we have 𝒟𝒟 𝑇𝑇 = {0} we also have Θ 𝑇𝑇 = 0. That is, Θ 𝑇𝑇 = 0 ∈ 𝐵𝐵(𝒟𝒟 𝑇𝑇 , 𝒟𝒟 𝑇𝑇 * ) = 𝐵𝐵({0}, 𝐾𝐾𝐾𝐾𝐾𝐾(𝑇𝑇)). Since by the von Neumann-Wold decomposition an isometry is a direct sum of a unitary and a unilateral shift, the result follows.

Remark. Note that in the proof of Theorem 2.3 Θ 𝑇𝑇 is constant since Θ 𝑇𝑇 (0) = Θ 𝑇𝑇 (𝜆𝜆). From Theorem 2.3 we conclude that if 𝑇𝑇 is an isometry, then Θ 𝑇𝑇 (0) = Θ 𝑇𝑇 (𝜆𝜆), for 𝜆𝜆 ∈ 𝔻𝔻. That is, Θ 𝑇𝑇 is constant.

Theorem 2. 4 Let 𝑇𝑇 ∈ 𝐵𝐵(ℋ). Then 𝛩𝛩 𝑘𝑘𝑇𝑇 (𝜆𝜆) = 𝑘𝑘𝛩𝛩 𝑇𝑇 (𝑘𝑘𝜆𝜆), 𝜆𝜆 ∈ 𝔻𝔻 holds for any 𝑘𝑘 ∈ 𝜕𝜕𝔻𝔻.

Proof. By a simple computation and using the definition we have

Since 𝑘𝑘 ∈ 𝜕𝜕𝔻𝔻, 𝑘𝑘 = 𝐾𝐾 𝑖𝑖𝑛𝑛𝑖𝑖 , 0 ≤ 𝜃𝜃 ≤ 2𝜋𝜋 and 𝑅𝑅 = 0, ±1, ±2, . .. Thus 𝑘𝑘𝑘𝑘 = 1. Thus the previous equation becomes

Once again, since 𝑘𝑘 ∈ 𝜕𝜕𝔻𝔻, 𝑘𝑘𝑘𝑘 = 1. Thus, the above equation simplifies to

This completes the proof. Theorem 2.5 [5] Two c.n.u contractions T 1 and T 2 are unitarily equivalent if and only if their characteristic functions coincide.

Proof. Suppose 𝑇𝑇 1 and 𝑇𝑇 2 are unitarily equivalent c.n.u contractions. Then there exists a unitary operator 𝑈𝑈 such that 𝑇𝑇 1 = 𝑈𝑈 * 𝑇𝑇 2 𝑈𝑈. Using the definition of the characteristic function and the fact that we have Without loss of generality, we let 𝑉𝑉 = 𝑈𝑈 * . The proof of the converse is trivial.

The above result can also be proved using the Neumann series and some computation as follows:

for some unitary operator 𝑈𝑈 and a c.n.u. contraction 𝑇𝑇. This result also says that if 𝑇𝑇 and 𝑆𝑆 are unitarily equivalent c.n.u contractions, then their characteristic functions Θ 𝑇𝑇 and Θ 𝑆𝑆 are unitarily equivalent, with the unitary equivalence being implemented by the same operator 𝑈𝑈.

Remark: Theorem 2.5 shows that the converse to Theorem 2.2 holds for c.n.u. contractions. Theorem 2.5 says that the characteristic function, modulo coincidence, is a complete unitary invariant for c.n.u contractions. This indicates that it should be possible to recover a c.n.u contraction, up to unitary equivalence, from its characteristic function.

Recall that a 𝐶𝐶 11 contraction is quasisimilar to a unitary operator and not similar. The following result was proved by Kerchy [3] .

Theorem 2.6 If T is a c.n.u contraction and the characteristic function Θ T of T is constant, then T is the orthogonal direct sum of a unilateral shift and a C 11 contraction.

Theorem 2.6 says that a c.n.u contraction with a constant characteristic function decomposes as a direct sum of a unilateral shift and an operator quasisimilar to a unitary operator. But since 𝑇𝑇 is c.n.u, the 𝐶𝐶 11 part of 𝑇𝑇 cannot be unitary, otherwise this would contradict the complete non-unitarity of 𝑇𝑇. The following result was proved by Nagy and Foia𝑠𝑠 [5] .

Theorem 2.7 [5] , (Theorem I.3.2). To every contraction T on a Hilbert space ℋ there corresponds a uniquely determined decomposition of ℋ into an orthogonal sum of subspaces reducing T, say ℋ = ℳ 1 ⊕ ℋ 1 , such that T 0 = T| ℳ 1 is unitary and T 1 = T| ℋ 1 is c.n.u. In particular, for an isometry, this canonical decomposition coincides with the von Neumann-Wold decomposition.

For a contraction 𝑇𝑇 with decomposition 𝑇𝑇 = 𝑇𝑇 0 ⊕ 𝑇𝑇 1 as in Theorem 2.7, we have 𝐷𝐷 𝑇𝑇 = 0 ⊕ 𝐷𝐷 𝑇𝑇 1 , 𝐷𝐷 𝑇𝑇 * = 0 ⊕ 𝐷𝐷 𝑇𝑇 1 * , 𝒟𝒟 𝑇𝑇 = 𝒟𝒟 𝑇𝑇 1 and 𝒟𝒟 𝑇𝑇 * = 𝒟𝒟 𝑇𝑇 1 * . These results lead us to the following result.

We present a simple method to show that Θ 𝑇𝑇 (𝜆𝜆) = Θ 𝑇𝑇 1 (𝜆𝜆), using the notion of orthogonal projections from ℋ = ℳ 0 ⊕ ℳ 1 onto ℳ 0 and ℳ 1 , respectively.

Corollary 2.8 Let 𝑇𝑇 ∈ 𝐵𝐵(ℋ) have the decomposition 𝑇𝑇 = 𝑇𝑇 0 𝑇𝑇 1 , with respect to the direct sum decomposition ℋ = ℳ 0 ⊕ ℳ 1 , where 𝑇𝑇 0 is unitary and 𝑇𝑇 1 is c.n.u. Then 𝛩𝛩 𝑇𝑇 (𝜆𝜆) = 𝛩𝛩 𝑇𝑇 1 (𝜆𝜆).

Proof. First, we note that 𝑇𝑇 = 𝑇𝑇 0 ⊕ 𝑇𝑇 1 = 𝑇𝑇 0 𝑃𝑃 0 + 𝑇𝑇 1 𝑃𝑃 1 , where 𝑃𝑃 0 and 𝑃𝑃 1 are the orthogonal projections from ℋ onto ℳ 0 and ℳ 1 , respectively. Clearly 𝐷𝐷 𝑇𝑇 = (𝐼𝐼 -𝑇𝑇 1 * 𝑇𝑇 1 )𝑃𝑃 1 and 𝐷𝐷 𝑇𝑇 * = (𝐼𝐼 -𝑇𝑇 1 * 𝑇𝑇 1 )𝑃𝑃 1 , and consequently, 𝐷𝐷 𝑇𝑇 = 𝐷𝐷 𝑇𝑇 1 𝑃𝑃 1 and 𝐷𝐷 𝑇𝑇 * = 𝐷𝐷 𝑇𝑇 1 * 𝑃𝑃 1 . This leads to the fact that 𝒟𝒟 𝑇𝑇 = 𝒟𝒟 𝑇𝑇 1 ⊆ ℳ 1 and 𝒟𝒟 𝑇𝑇 * = 𝒟𝒟 𝑇𝑇 1 * ⊆ ℳ 1 . A simple computation and an application of Theorem 2.3 using the fact that 𝑇𝑇 0 | 𝒟𝒟 𝑇𝑇 1 = 0 and 𝑇𝑇 0 * 𝑛𝑛-1 | 𝒟𝒟 𝑇𝑇 1 = 0 we get Theorem 2.9 If 𝑇𝑇 is a contraction, then 𝛩𝛩 𝑇𝑇 (𝜆𝜆) is a unitary operator on any arc of 𝜕𝜕𝔻𝔻 belonging to the resolvent set of 𝑇𝑇.

Proof. Denote by Ω 𝑇𝑇 the set of complex numbers 𝜆𝜆 for which the operator (𝐼𝐼 -𝜆𝜆𝑇𝑇 * ) is boundedly invertible. For 𝜆𝜆 ∈ Ω 𝑇𝑇 and using the Neumann series for (𝐼𝐼 -𝜆𝜆𝑇𝑇 * ) -1 , we define

Clearly Ω 𝑇𝑇 is an open set that contains 𝔻𝔻 and Θ 𝑇𝑇 (𝜆𝜆) is analytic on Ω 𝑇𝑇 . Applying (2.1) to 𝑇𝑇 * as well as to 𝑇𝑇, we obtain

𝜆𝜆 is a point on the unit circle) belonging to the resolvent set 𝜌𝜌(𝑇𝑇), then 𝜆𝜆 = (𝜆𝜆) -1 ∈ Ω 𝑇𝑇 and hence

This proves that Θ 𝑇𝑇 (𝜆𝜆) is a unitary operator. The class of nilpotent contractions yields a natural set of examples of c.n.u contractions with polynomial characteristic functions. For example, let 𝑇𝑇 be a contraction and nilpotent of order 𝑅𝑅, where 𝑅𝑅 ≥ 1. That is, ∥ 𝑇𝑇 ∥≤ 1 and 𝑇𝑇 𝑛𝑛 = 0 and 𝑇𝑇 𝑛𝑛-1 ≠ 0. Therefore 𝐷𝐷 𝑇𝑇 * 𝑇𝑇 * 𝑘𝑘 𝐷𝐷 𝑇𝑇 = 0 for all 𝑘𝑘 ≥ 𝑅𝑅. Using this fact, the characteristic function Θ 𝑇𝑇 of 𝑇𝑇 is given by

, for all 𝜆𝜆 ∈ 𝔻𝔻 (see) [2] . This shows that Θ 𝑇𝑇 is a polynomial of degree less or equal to 𝑅𝑅.

Structure of c.n.u. Contractions

Recall that a pure isometry is a unilateral shift of some multiplicity and a pure co-isometry is the adjoint of a pure isometry. It has been shown in [2] that the characteristic function Θ 𝑇𝑇 of a c.n.u. contraction 𝑇𝑇 on a Hilbert space ℋ is a polynomial of degree ≤ 𝑅𝑅 if and only if there exist three subspaces ℋ 1 , ℳ 1 , ℋ -1 of ℋ such that ℋ has the orthogonal decomposition ℋ = ℋ 1 ⊕ ℳ 1 ⊕ ℋ -1 and a pure isometry 𝑆𝑆 ∈ 𝐵𝐵(ℋ 1 ), a nilpotent operator 𝑁𝑁 ∈ 𝐵𝐵(ℳ 1 ) of order 𝑅𝑅 and a pure co-isometry 𝐶𝐶 ∈ 𝐵𝐵(ℋ -1 ) such that 𝑇𝑇 has the following matrix representation or triangulation 𝑇𝑇 = � 𝑆𝑆 * * 0 𝑁𝑁 * 0 0 𝐶𝐶 �. The spaces on which one or two of the diagonal blocks acts may be {0} and this leads to the following six degenerate cases:

Function models for the degenerate cases of contractions have been constructed in [2] . The case when 𝑇𝑇 = [𝑆𝑆] has been solved in Theorem 2. In Theorem 2.10, 𝐶𝐶 is a co-isometry. The spaces on which the two diagonal blocks acts may be the zero space {0}. Therefore some of these entries may be absent. If 𝑇𝑇 is an isometry, then the characteristic function is a constant. In this particular case, Θ 𝑇𝑇 (𝜆𝜆) and Θ 𝑇𝑇 (0) not only coincide but are equal, this also applies to a unilateral shift, since it is an isometry.

We give an examples of the case when 𝑇𝑇 = [𝑁𝑁], where 𝑁𝑁 has order 2,3 in ℂ 2 and ℂ 3 .

Example 2.11

For T = � 0 1 0 0 � acting on the Hilbert space ℋ = ℂ 2 , it is easy to show that the 2 × 2-matrix-valued characteristic

�} . This shows that Θ T is a polynomial of degree less or equal to 2. A simple computation also shows that

which agrees with Theorem 2.9. We note also that Θ 𝑇𝑇 (𝜆𝜆) -1 is not contractive for 𝜆𝜆 → 0. This means that ∥ Θ 𝑇𝑇 (𝜆𝜆) -1 𝐴𝐴 ∥≥∥ 𝐴𝐴 ∥ for all 𝐴𝐴 ≠ 0 as 𝜆𝜆 → 0.

The following example gives a c.n.u contraction which is nilpotent of order 3 but it has a constant characteristic polynomial.

Consider 𝑇𝑇 = � 0 1 0 0 0 1 0 0 0 �. A simple computation shows that Θ 𝑇𝑇 (𝜆𝜆) = Θ 𝑇𝑇 (0) = -𝑇𝑇| 𝒟𝒟 𝑇𝑇 . This shows that the characteristic function is independent of 𝜆𝜆 and is thus constant. That is, it is of degree zero.

Note that the characteristic function of these degenerate cases may be independent of the order of 𝑁𝑁. For instance, the operator

A contractive analytic function Θ: 𝔻𝔻 ⟶ 𝐵𝐵(𝒟𝒟 𝑇𝑇 , 𝒟𝒟 𝑇𝑇 * ) is called inner if its boundary values Θ(𝐾𝐾 𝑖𝑖𝑖𝑖 ) are isometries (or have modulus 1) almost everywhere on 𝜕𝜕𝔻𝔻.

Theorem 2.12 Let 𝑇𝑇 ∈ 𝐵𝐵(ℋ) be a c.n.u contraction. Then 𝑇𝑇 ∈ 𝐶𝐶 .0 if and only if 𝛩𝛩 𝑇𝑇 is inner.

A c.n.u. contraction 𝑇𝑇 such that Ψ(𝑇𝑇) = 0, for some nonzero function Ψ ∈ ℍ ∞ , then 𝑇𝑇 𝑛𝑛 ⟶ 0 and 𝑇𝑇 * 𝑛𝑛 ⟶ 0, i.e. 𝑇𝑇 ∈ 𝐶𝐶 00 . Thus a c.n.u contraction 𝑇𝑇 ∈ 𝐶𝐶 00 is said to be a 𝐶𝐶 0 -contraction if there exists an inner function Ψ such that Ψ(𝑇𝑇) = 0.

Therefore, taking the 𝒟𝒟 𝑇𝑇 * -norm both sides and using the Pythagoras Theorem and the fact that 𝑇𝑇 is pure, we have

This proves that 𝑈𝑈 is an isometry, and hence injective. It is not easy to show that 𝑅𝑅𝑅𝑅𝑅𝑅(𝑈𝑈) = 𝒟𝒟 𝑇𝑇 * . However, it is easy to show that 𝑇𝑇 is invertible. This follows from the fact that since 𝑇𝑇 is a pure contraction, so is 𝑇𝑇 * (since ∥ 𝑇𝑇 * ∥=∥ 𝑇𝑇 ∥< 1) and so 𝐷𝐷 𝑇𝑇 * is invertible. By hypothesis (𝐼𝐼 ℋ -𝜆𝜆𝑇𝑇 * ) -1 is also invertible, which means that 𝑈𝑈 is an invertible isometry. This establishes the claim.

Example. For the operator 𝑇𝑇 = � 0 1 0 0 �, a simple calculation gives that |𝑇𝑇| = � 0 0 0 1 �, and

We note that there is a relationship between 𝑇𝑇 and Θ 𝑇𝑇 (𝜆𝜆). In fact, sup{∥ Θ 𝑇𝑇 (𝜆𝜆) ∥: ∥ 𝜆𝜆 ∥< 1} = 𝐶𝐶 =∥ 𝑇𝑇 ∥. ).

Proof. The first claim is straightforward from definition. For the second claim, we first note that Proof. Obvious from the definition of Θ 𝑇𝑇 .

A unilateral shift is the classical example of non-unitary isometry. In fact, they are prototypes of non-unitary isometries. Any truncated shift operator is nilpotent.

Theorem 3.7 [4] (Problem 5.3) Two shifts are unitarily equivalent if and only if they have the same multiplicity.

The following result is a consequence of Theorem 3.7.

Corollary 3.8 Two c.n.u contractions have same multiplicity if and only if their characteristic functions coincide.

Discussion

Contractive operator-valued analytic functions play an important role in operator theory and serve as a bridge between operator theory and function theory in terms of system realization theory, interpolation theory and scattering and transfer function theory in classical and quantum physics and linear time invariant systems (cf.) [1,5,6] . Clearly a contraction operator can be investigated in details in terms of its Sz.-Nagy-Foia𝑠𝑠 characteristic function.

Conclusion

This study explores key properties and behaviors of contractions in Hilbert spaces, focusing on their characteristic functions and decompositions. Through the application of Sz.-Nagy-Foias theory, the work demonstrates that completely non-unitary (c.n.u.)

Acknowledgement

We call 𝐶𝐶 0 the class of those completely non-unitary contractions 𝑇𝑇 for which there exists a nonzero function Ψ ∈ ℍ ∞ such that Ψ(𝑇𝑇) = 0. Clearly 𝐶𝐶 0 ⊂ 𝐶𝐶 00 . In Nagy and Foias [5] it is observed that for 𝑇𝑇 ∈ 𝐶𝐶 0 , the function Ψ in the definition can always be taken to be inner. Clearly Θ 𝑇𝑇 is inner precisely when 𝑇𝑇 𝑛𝑛 ⟶ 0 strongly(i.e. 𝑇𝑇 ∈ 𝐶𝐶 0. ). AcknowledgementsThe authors sincerely wish to thank referees for useful comments and suggestions during the writing of this manuscript. The authors also pay tribute to the Eastern Africa Universities Mathematics Programme(EAUMP)/ICTP/ISP and the University of Nairobi for funding this research.

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