Minimum attaining operators on reducing subspaces: Spectral structure and density

Abstract: In this article, we introduce and investigate a new subclass $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$. This class contains the absolutely minimum attaining operators and is properly contained in the class of minimum attaining operators. We establish several structural and spectral characterizations of operators in $\mathcal{M}_r(H)$. In particular, we characterize positive operators in $\mathcal{M}_r(H)$ in terms of their spectral representations. We prove that $\mathcal{M}_r(H)$ is dense in $\mathcal{B}(H)$ in the operator norm and, moreover, that the operators in $\mathcal{M}_r(H)$ having a nontrivial invariant half-space are also dense in $\mathcal{B}(H)$ in the operator norm. We further obtain a representation theorem for normal operators in $\mathcal{M}_r(H)$ and establish additional structural properties of this class.

• The papers defines the class of operators $ \mathcal{M}_{mul}(H) of bounded operators that attain their minimum modulus on every reducing subspace.
• Self-adjoint operators in $\mathcal{ M}_{mul}(H)$ have a well-ordered point spectrum and are diagonalizable
• The class $\mathcal{ M}_{mul}(H)$ is dense in the space of all bounded operators, indicating prevalence in operator theory.

The intermediate class $\mathcal{M}_r(H)$

This paper introduces and develops the theory of operators that attain their minimum modulus on every nonzero reducing subspace. For a bounded operator $T$ on a separable complex Hilbert space $H$, the minimum modulus is $m(T)=\inf\{\|Tx\|:x\in S_H\}$, and $T$ is minimum attaining if this infimum is realized on the unit sphere. The authors define

$$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$

where $\mathcal{R}_T$ denotes the lattice of reducing subspaces of $T$. This class interpolates between the absolutely minimum attaining operators $\mathcal{AM}(H)$ and the minimum attaining operators $\mathcal{M}(H)$:

$T$0

Both inclusions are shown to be strict. Strictness on the left follows because every idempotent lies in $T$1, while an idempotent with both range and kernel infinite dimensional fails to be AM (by a known characterization of AM idempotents). Strictness on the right follows from an explicit diagonal operator on $T$2 that attains its minimum at $T$3 but whose restriction to $T$4 has minimum modulus $T$5 that is not an eigenvalue. The definition parallels the $T$6-class of norm attaining operators studied by Ramesh and Osaka, where norm attainment is required on every reducing subspace; here all results are proved independently using the minimum-attainment property.

The paper also identifies broad natural subclasses contained in $T$7: every irreducible minimum attaining operator, all idempotents, orthogonal projections, partial isometries, Toeplitz operators with inner symbols, and Hankel operators $T$8 with $T$9 inner. Membership is invariant under unitary equivalence, and for WEP operators ($H$0), membership of $H$1 is equivalent to membership of $H$2. In general, however, $H$3 does not imply $H$4 or $H$5; a unilateral weighted shift example demonstrates all three failures simultaneously, including non-closed range.

A useful reduction is that $H$6 implies $H$7, since $H$8 and $H$9; for positive $m(T)=\inf\{\|Tx\|:x\in S_H\}$0, membership of $m(T)=\inf\{\|Tx\|:x\in S_H\}$1 is equivalent to that of $m(T)=\inf\{\|Tx\|:x\in S_H\}$2.

Spectral structure: well-ordered point spectra

The central spectral result concerns self-adjoint operators: if $m(T)=\inf\{\|Tx\|:x\in S_H\}$3 is self-adjoint, then $m(T)=\inf\{\|Tx\|:x\in S_H\}$4 admits an orthonormal basis of eigenvectors, and the set $m(T)=\inf\{\|Tx\|:x\in S_H\}$5 is a countable well-ordered subset of $m(T)=\inf\{\|Tx\|:x\in S_H\}$6. Consequently $m(T)=\inf\{\|Tx\|:x\in S_H\}$7 admits the strong-operator-topology representation $m(T)=\inf\{\|Tx\|:x\in S_H\}$8 over mutually orthogonal eigenprojections. The proof of diagonalizability is a Zorn's lemma argument: a maximal orthonormal set of eigenvectors spans a reducing subspace, and minimum attainment forces an eigenvector in any nonzero complement. Well-ordering follows from applying the defining condition to the reducing subspace spanned by eigenspaces indexed by an arbitrary subset $m(T)=\inf\{\|Tx\|:x\in S_H\}$9 of $T$0: minimum attainment forces $T$1 itself to lie in $T$2.

The mechanism behind well-ordering is transparent: a positive operator with strictly decreasing eigenvalues $T$3 fails membership, since the direct sum of the corresponding eigenspaces is reducing but the infimum $T$4 is not attained there. Combining necessity and sufficiency yields a complete characterization:

Characterization: A positive operator $T$5 belongs to $T$6 if and only if $T$7, where $T$8 is a bounded countable well-ordered set and $T$9 are mutually orthogonal projections summing to $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$0 strongly. In particular $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$1, so the spectrum of such an operator is countable with at most one accumulation point, namely its infimum.

There is also a clean bridge to the $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$2-class: for positive $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$3 with $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$4, one has $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$5 if and only if $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$6. This affine correspondence reflects that, on positive operators, minimum attainment of $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$7 coincides with norm attainment of its reflection about $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$8.

Stability under perturbations and density

Two perturbation results show robustness of the class. First, adding a self-adjoint finite-rank operator commuting with each $$\mathcal{M}_r(H):=\{T\in\mathcal{B}(H): T|_M\in\mathcal{M}(M)\ \text{for every nonzero } M\in\mathcal{R}_T\},$$9 to a well-ordered spectral sum preserves membership, because finitely many points are removed from and adjoined to a well-ordered set. Second, subtracting a positive compact operator $\mathcal{R}_T$0 commuting with the $\mathcal{R}_T$1 and dominated by $\mathcal{R}_T$2 on each range $\mathcal{R}_T$3 also preserves membership; the proof shows that any putative decreasing sequence in $\mathcal{R}_T$4 would force either infinitely many increasing nonzero eigenvalues of a compact operator or a contradiction via compactness applied to an orthonormal sequence. These combine into a theorem covering $\mathcal{R}_T$5 perturbations of positive type, which in turn yields that finite direct sums of AM operators lie in $\mathcal{R}_T$6. Notably, this fails for infinite direct sums: $\mathcal{R}_T$7 has point spectrum without a least element and hence is not in the class.

The density results are the strongest quantitative statements in the paper:

$\mathcal{R}_T$8

The argument approximates $\mathcal{R}_T$9 on $T$0 by a finite-spectrum self-adjoint operator, forms $T$1 via the polar decomposition, notes $T$2 has finite spectrum and thus lies in the class, and concludes $T$3. Moreover, refining with a rank-one-type perturbation technique of Tcaciuc, the authors prove that the operators in $T$4 possessing a nontrivial invariant half-space are also norm dense in $T$5 — a strengthening relevant to the invariant subspace problem, since it shows dense subsets of $T$6 consist of operators guaranteed to have invariant subspaces of infinite dimension and codimension. Since $T$7, these density theorems strictly improve the known density of $T$8 due to Kulkarni and Ramesh.

Normal and multiplication operators

For quasinormal $T$9, membership of $\mathcal{AM}(H)$0 is equivalent to $\mathcal{AM}(H)$1; the proof shows quasinormality makes the closed span of eigenspaces of $\mathcal{AM}(H)$2 reduce $\mathcal{AM}(H)$3, allowing the Zorn-type argument to run for $\mathcal{AM}(H)$4 directly. For normal operators this yields a full structural representation: $\mathcal{AM}(H)$5 if and only if $\mathcal{AM}(H)$6 decomposes as an orthogonal direct sum of pairwise orthogonal reducing subspaces $\mathcal{AM}(H)$7, with $\mathcal{AM}(H)$8 countable well-ordered, such that

$\mathcal{AM}(H)$9

where each $\mathcal{M}(H)$0 is unitary (with $\mathcal{M}(H)$1 arbitrary on $\mathcal{M}(H)$2). Three consequences follow immediately: normal members have adjoints in the class, normal members have closed range (since $\mathcal{M}(H)$3 is injective and minimum attaining, hence bounded below), and multiplying a normal member on either side by an isometry preserves membership — the last requiring a careful case analysis when $\mathcal{M}(H)$4.

As a concrete application, multiplication operators on $\mathcal{M}(H)$5 are characterized symbol-theoretically: $\mathcal{M}(H)$6 if and only if

$\mathcal{M}(H)$7

with $\mathcal{M}(H)$8 unimodular, $\mathcal{M}(H)$9 pairwise essentially disjoint measurable sets of positive measure covering $T$00 up to null sets, and $T$01 a countable well-ordered subset of $T$02. Thus $T$03 qualifies exactly when $T$04 takes countably many values indexed by a well-ordered set.

General operators and the hypothesis $T$05

Without normality or quasinormality, $T$06 and $T$07 need not share reducing subspaces. Under the structural assumption $T$08, the equivalence $T$09 holds verbatim, and the paper obtains a complete representation: $T$10 belongs to the class if and only if

$T$11

where $T$12 is countable well-ordered, $T$13 are mutually orthogonal projections summing to $T$14, and $T$15 are partial isometries with initial spaces $T$16 satisfying $T$17; convergence is in the strong operator topology. This is obtained from the polar decomposition $T$18 by restricting the partial isometry to each spectral subspace.

Finally, the hypothesis itself is characterized through von Neumann algebras: writing $T$19, one has $T$20 if and only if the partial isometry $T$21 belongs to the bicommutant $T$22. The forward direction shows $T$23 commutes with every projection in $T$24 (handling $T$25 separately), then extends to the whole commutant by strong density of projections; the converse uses commutation with spectral projections of $T$26 to transfer reducibility. The authors explicitly pose as an open question the search for intrinsic necessary and sufficient conditions for $T$27 beyond this algebraic criterion.

Limitations and open questions

Several restrictions qualify the results. Membership is not preserved under taking adjoints or moduli in general, so much of the theory relies on auxiliary hypotheses (self-adjointness, quasinormality, normality, WEP, or $T$28). The spectral characterizations apply directly only to positive and self-adjoint operators; for arbitrary $T$29 no spectral description independent of the structural hypothesis is given. Infinite direct sums of AM operators fall outside the class, so the finite-sum stability result cannot be extended naively. The paper leaves open the characterization of $T$30 stated above, and the analogous question of whether the compact-perturbation results extend to broader classes of commuting perturbations is not addressed.

Conclusion

The paper establishes $T$31 as a properly intermediate class between absolutely minimum attaining and minimum attaining operators, gives complete spectral characterizations in the positive, normal, and structurally constrained settings via countable well-ordered spectra, proves norm denseness of the class — and of its subclass with nontrivial invariant half-spaces — in $T$32, and links membership for normal and multiplication operators to concrete decompositions. The remaining gap between the general theory and the cases governed by $T$33 defines the natural continuation of this line of work.