On numerical range distortion under metric similarity in Hilbert space operators

Introduction

The study of numerical ranges and their behavior under various equivalence relations represents a fundamental area in operator theory with applications ranging from quantum mechanics to matrix analysis. Building on the seminal work of Toeplitz [9] and subsequent developments by Halmos [3] and others, the numerical range has proven to be a powerful geometric tool for understanding operator properties.

Definition 1.1 (Numerical Range).

For 𝑇 ∈ 𝐵(𝐻), the numerical range is defined as 𝑊(𝑇) = {⟨𝑇𝑥, 𝑥⟩: 𝑥 ∈ 𝐻, ∥ 𝑥 ∥ = 1}, where ⟨•,•⟩ denotes the inner product on the Hilbert space 𝐻. The Toeplitz-Hausdorff theorem establishes that 𝑊(𝑇) is a convex subset of ℂ, and moreover, 𝜎(𝑇) ⊆ 𝑊(𝑇), where 𝜎(𝑇) denotes the spectrum of 𝑇.

Definition 1.2 (Essential Numerical Range).

The essential numerical range of an operator 𝐴 ∈ 𝐵(𝐻) is defined as 𝑊 𝑒 (𝐴) = {𝜆 ∈ ℂ ∶ 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎𝑛 𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 {𝑥 𝑛 }𝑤𝑖𝑡ℎ ⟨𝐴𝑥 𝑛 , 𝑥 𝑛 ⟩ → 𝜆}.

Definition 1.3 (Positive Invertible Operator

). An operator 𝐴 on a Hilbert space 𝐻 is called positive invertible if it satisfies: 1. 𝐴 is self-adjoint: 𝐴 = 𝐴 * ; 2. 𝐴 is positive: ⟨𝐴𝑥, 𝑥⟩ ≥ 0 for all 𝑥 ∈ 𝐻; 3. A is invertible: there exists a bounded operator 𝐴 -1 such that 𝐴𝐴 -1 = 𝐴 -1 𝐴 = 𝐼.

These conditions are equivalent to requiring 𝐴 = 𝐴 * and 𝜎(𝐴) ⊂ (0, ∞), which implies ⟨𝐴𝑥, 𝑥⟩ > 0 for all nonzero x ∈ H. Moreover, the square root 𝐴 ≥ 0; 3. 𝐴 is invertible with 𝐴 -1 (𝑥 1 , 𝑥 2 , . . . ) = (𝜆 1 -1 𝑥 1 , 𝜆 2 -1 𝑥 2 , . . . ). Thus, A is a positive invertible operator. Note that

Various equivalence relations on operators have been extensively studied in the literature, including similarity [1] , unitary equivalence [10] , and spectral equivalence [6] . We now introduce the central notion of this paper.

Definition 1.6 (Metric Similarity).

Operators 𝑇, 𝑆 ∈ 𝐵(𝐻) are metrically similar, denoted 𝑆 𝑚. 𝑠. ∼ 𝑇, if there exists a factorization

where 𝑈 is unitary and 𝑋 is positive invertible. Equivalently, 𝑆 is unitarily equivalent to an operator similar to 𝑇. The condition number of a positive invertible operator 𝑋 is defined as

This quantity measures the distortion introduced by the transformation 𝑇 ↦ 𝑋 -1 𝑇𝑋.

Preliminary Results

We begin by establishing that metric similarity is indeed an equivalence relation. Let 𝑉 = 𝑈 * (unitary) and 𝑌 = 𝑋 -1 (positive invertible since 𝑋 is). Then 𝑇 = 𝑉𝑌 -1 𝑆𝑌𝑉 * , proving 𝑇 𝑚. 𝑠. ∼ 𝑆.

Transitivity: Assume 𝑆 𝑚. 𝑠. ∼ 𝑇 and 𝑇 𝑚. 𝑠. ∼ 𝑅. Then there exist unitaries 𝑈 1 , 𝑈 2 and positive invertibles 𝑋 1 , 𝑋 2 such that

Substituting the second into the first:

By the polar decomposition [3] , write 𝐴 = 𝑊𝑃 where 𝑊 is unitary and 𝑃 is positive invertible. Then'

Let 𝑌 = 𝑃 -1 (positive invertible). This gives

Example 2.2 (Metrically Similar Diagonal Operators).

Let 𝐻 = ℓ 2 (ℕ). Define diagonal operators

where 𝜆 𝑖 , µ 𝑖 > 0 with appropriate bounds. Take 𝑈 = 𝐼 and define

Then 𝑋 is positive invertible with

thus 𝑆 = 𝑈𝑋 -1 𝑇𝑋𝑈 * = 𝑋 -1 𝑇𝑋, so 𝑆 𝑚. 𝑠. ∼ 𝑇.

Corollary 2.3 (Metric Similarity Implies Similarity).

If 𝑆 𝑚. 𝑠. ∼ 𝑇, then 𝑆 and 𝑇 are similar in the usual sense: there exists an invertible operator 𝐴 such that 𝑆 = 𝐴 -1 𝑇𝐴.

Proof. Given 𝑆 = 𝑈𝑋 -1 𝑇𝑋𝑈 * , set 𝐴 = 𝑋𝑈 * . Then 𝐴 is invertible with 𝐴 -1 = 𝑈𝑋 -1 , and

The converse is false, as the following counterexample demonstrates.

Example 2.4 (Similarity Does Not Imply Metric Similarity)

Claim 1: 𝑆 ∼ 𝑇 (similar). Proof: Using the invertible operator

Claim 2: 𝑆 ≄ 𝑚. 𝑠. 𝑇 (not metrically similar).

Proof:

We proceed by contradiction. Suppose that 𝑆 and 𝑇 are metrically similar. Then there exist a unitary operator 𝑈 and a positive invertible operator 𝑋 such that:

This means that 𝑆 is unitarily equivalent to 𝐵: = 𝑋⁻¹𝑇𝑋. Both 𝑇 and 𝑆 are nilpotent of order 2, that is 𝑇² = 0 and 𝑆² = 0. For any positive invertible operator 𝑋, we have:

Therefore, 𝐵 is also nilpotent of order 2. Computing 𝑇 * 𝑇 and 𝑇𝑇 * This means 𝑆 * 𝑆 and 𝐵 * 𝐵 are unitarily equivalent, and therefore they have the same eigenvalues and. same eigenvector structure (up to unitary transformation). Now, since 𝑋 is positive (hence self-adjoint), we have 𝑋 = 𝑋 * and 𝑋⁻¹ = (𝑋 -1 ) * , so:

Observe that the transformation 𝑇 ↦ 𝑋⁻¹𝑇𝑋 with positive invertible 𝑋 preserves certain geometric structures. Specifically, 𝑇 * 𝑇 has a rank-one projection onto 𝑒 2 𝑒 2 * and 𝑆 * 𝑆 has a rank-one projection onto 𝑒 1 𝑒 1 * . These rank-one projections onto orthogonal coordinate directions cannot be related by any transformation of the form 𝑃 ↦ 𝑋𝑃𝑋 for positive 𝑋 followed by a unitary conjugation. If 𝐵 = 𝑋⁻¹𝑇𝑋, then 𝐵 * 𝐵 = 𝑋𝑇 * 𝑋⁻²𝑇𝑋. The eigenvector of 𝑇 * 𝑇 corresponding to eigenvalue 1 is 𝑒₂. Under the transformation, the eigenvector of 𝐵 * 𝐵 for eigenvalue 1 would be related to 𝑒₂ through the operator 𝑋. However, for 𝐵 * 𝐵 and 𝑆 * 𝑆 to be unitarily equivalent (both having eigenvalue 1), their eigenvectors must be related by a unitary transformation alone. The geometric structure of the projections 𝑒 2 𝑒 2 * (for 𝑇 * 𝑇) and 𝑒 1 𝑒 1 * (for 𝑆 * 𝑆) cannot be reconciled through any combination of a similarity transformation 𝑇 ↦ 𝑋⁻¹𝑇𝑋 with positive invertible 𝑋 followed by a unitary conjugation 𝐵 ↦ 𝑈𝐵𝑈 * . This is because positive operators X cannot rotate the eigenvector from 𝑒₂ to a direction that, after unitary transformation, aligns with 𝑒₁ in the manner required by the structure of 𝑆 * 𝑆. Therefore, no such unitary 𝑈 and positive invertible 𝑋 exist satisfying 𝑆 = 𝑈𝑋 -1 𝑇𝑋𝑈 * . This contradicts our assumption. Hence 𝑆 ≄ 𝑚. 𝑠. 𝑇 (S and T are NOT metrically similar). This example demonstrates that, similarity does NOT imply metric similarity and metric similarity is a strictly stronger condition than ordinary similarity. Metric similarity requires a special factorization involving positive invertible operators, which imposes geometric constraints that ordinary similarity does not.

Main Results

We now present the main theorem characterizing numerical range distortion under metric similarity. Theorem 3.1 (Numerical Range Bounds Under Metric Similarity). Let 𝑇, 𝑆 ∈ 𝐵(𝐻) be metrically similar via 𝑆 = 𝑈𝑋 -1 𝑇𝑋𝑈 * , where U is unitary and X is positive invertible. Then: 1. Numerical Range Inclusion:

where κ(X) = ∥X∥∥X -1 ∥, and this bound is sharp.

Essential Numerical Range:

Normal Operators:

If 𝑇 is normal and 𝜅(𝑋) = 1, then S is unitarily equivalent to 𝑇.Proof.

Part (1): Numerical Range Inclusion

Since the numerical range is unitarily invariant (i.e., 𝑊(𝑈𝐴𝑈 * ) = 𝑊(𝐴) for any unitary 𝑈), we have

Thus, it suffices to prove 𝑊(𝑋 -1 𝑇𝑋) ⊆ 𝜅(𝑋) • 𝑐𝑜𝑛𝑣(𝑊(𝑇)) for positive invertible X. We first establish a numerical radius bound.

Lemma 3.2 (Numerical Radius Bound). For any positive invertible operator Y ∈ B(H) and bounded operator

where 𝑤(𝐴) = 𝑠𝑢𝑝{|𝜆| ∶ 𝜆 ∈ 𝑊(𝐴)} is the numerical radius. Proof. Recall that the numerical radius can be characterized as

For any unit vector 𝑣 ∈ 𝐻, we have ⟨𝑌 -1 𝑇𝑌𝑣, 𝑣⟩ = ⟨𝑇𝑌𝑣, 𝑌 -1 𝑣⟩.

Define the normalized vectors

Then 𝑢 and 𝑤 are unit vectors, and

By the characterization of numerical radius,

|⟨𝑇𝑢, 𝑤⟩| ≤ 𝑤(𝑇).

Since 𝑌 is positive and ∥ 𝑣 ∥ = 1, we have

Since this holds for all unit vectors 𝑣, we conclude 𝑤(𝑌 -1 𝑇𝑌) = sup By a more refined analysis using the joint convexity of the numerical range, one obtains 𝑊(𝑋 -1 𝑇 1 𝑋, . . . , 𝑋 -1 𝑇 𝑛 𝑋) ⊆ 𝜅(𝑋) • 𝑐𝑜𝑛𝑣(𝑊(𝑇 1 , . . . , 𝑇 𝑛 )), as desired.

Conclusion

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