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arxiv: 2111.11713 · v3 · submitted 2021-11-23 · 🧮 math.FA · math.CV

Operator valued analogues of multidimensional Bohr's inequality

Vasudevarao Allu , Himadri Halder This is my paper

Pith reviewed 2026-05-24 12:54 UTC · model grok-4.3

classification 🧮 math.FA math.CV
keywords Bohr's inequalityoperator-valued functionsbounded analytic functionsmultidimensional inequalitiescomplete circular domainsHilbert space operators
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The pith

Bohr's inequality admits sharp improved versions for bounded analytic operator-valued functions that extend to multidimensional complete circular domains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes several sharp improved and refined versions of Bohr's inequality first for functions in the class of bounded analytic maps from the unit disk into the algebra of bounded operators on a Hilbert space. It then proves the multidimensional analogues of these operator-valued inequalities on complete circular domains in several complex variables. A reader would care because the classical Bohr inequality bounds the sum of absolute coefficient values by the supremum norm of the function, and the operator-valued extension preserves this control when coefficients become operators.

Core claim

For f in H^∞(D, B(H)), several sharp improved and refined versions of the Bohr inequality hold. For a complete circular domain Q subset C^n, the multidimensional analogues of the operator-valued Bohr inequality are established, together with the multidimensional analogues of several improved Bohr inequalities for operator-valued functions on Q.

What carries the argument

The class H^∞(D, B(H)) of bounded analytic functions from the unit disk into the algebra of bounded operators on a Hilbert space, together with its extension to complete circular domains Q.

If this is right

  • The classical scalar Bohr inequality is recovered as the special case when the Hilbert space is one-dimensional.
  • The improved coefficient bounds apply uniformly to power series whose coefficients are operators rather than scalars.
  • The multidimensional versions control the growth of multi-index coefficients for operator-valued functions on complete circular domains.
  • These inequalities remain valid for every complex Hilbert space H.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The results may supply coefficient estimates useful for studying operator-valued holomorphic maps on more general domains.
  • They suggest examining whether the same bounds persist when the target algebra is replaced by other operator algebras such as von Neumann algebras.

Load-bearing premise

The target space must be the algebra of bounded linear operators on a Hilbert space and the functions must be bounded and analytic.

What would settle it

An explicit bounded analytic function f from the unit disk to B(H) for which the sum of the operator norms of its Taylor coefficients exceeds the supremum norm bound stated in one of the claimed inequalities.

read the original abstract

Let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. In this paper, we first establish several sharp improved and refined versions of the Bohr's inequality for the functions in the class $H^{\infty}(\mathbb{D},\mathcal{B}(\mathcal{H}))$ of bounded analytic functions from the unit disk $\mathbb{D}:=\{z \in \mathbb{C}:|z|<1\}$ into $\mathcal{B}(\mathcal{H})$. For the complete circular domain $Q \subset \mathbb{C}^n$, we prove the multidimensional analogues of the operator valued Bohr's inequality established by G. Popescu [Adv. Math. 347 (2019), 1002-1053]. Finally, we establish the multidimensional analogues of several improved Bohr's inequalities for operator valued functions in $Q$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper establishes several sharp improved and refined versions of Bohr's inequality for functions in the class H^∞(D, B(H)) of bounded analytic operator-valued functions on the unit disk, and proves multidimensional analogues of Popescu's operator-valued Bohr inequality for complete circular domains Q ⊂ C^n, along with improved versions in that setting.

Significance. If the derivations hold, the work supplies explicit operator-valued extensions of Bohr-type inequalities with claimed sharpness, extending scalar results and Popescu's 2019 operator-valued work to the multivariable complete-circular case; this is a direct contribution to operator theory and several complex variables, with the explicit constants and refinements constituting the main technical advance.

major comments (2)
  1. [§3, Theorem 3.1] §3, Theorem 3.1: the claimed sharpness of the refined constant appears to rest on a specific choice of test function f(z) = zI; it is not shown whether this extremal is attained for general non-scalar operators in B(H), which would be needed to confirm the operator-valued refinement is optimal rather than merely an upper bound.
  2. [§4] §4, the passage from the one-variable case to the complete circular domain Q: the reduction step invokes the maximum-modulus principle on the polydisk but does not explicitly verify that the operator norm remains submultiplicative under the joint spectrum map when the variables fail to commute; a short calculation confirming this would strengthen the multidimensional claim.
minor comments (2)
  1. [Introduction] Notation: the symbol Q is introduced for the complete circular domain but its precise definition (e.g., whether it is balanced or Reinhardt) is only implicit; an explicit sentence in the introduction would aid readability.
  2. [Introduction] References: Popescu's 2019 paper is cited for the base operator-valued inequality, but the precise statement being extended (e.g., which theorem number) is not restated; adding a one-sentence recap would clarify the improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.1] the claimed sharpness of the refined constant appears to rest on a specific choice of test function f(z) = zI; it is not shown whether this extremal is attained for general non-scalar operators in B(H), which would be needed to confirm the operator-valued refinement is optimal rather than merely an upper bound.

    Authors: The sharpness is demonstrated by the test function f(z) = zI, which is a valid element of the class H^∞(D, B(H)) for any Hilbert space H. For this function, the refined inequality achieves the constant, establishing that it cannot be improved. This holds regardless of whether the operator is scalar or non-scalar, as zI is non-scalar when dim H > 1. The optimality is with respect to the class, not that every function attains equality. We believe the manuscript already correctly establishes the sharpness in the operator-valued context. revision: no

  2. Referee: [§4] the passage from the one-variable case to the complete circular domain Q: the reduction step invokes the maximum-modulus principle on the polydisk but does not explicitly verify that the operator norm remains submultiplicative under the joint spectrum map when the variables fail to commute; a short calculation confirming this would strengthen the multidimensional claim.

    Authors: We agree that an explicit verification would be helpful. In the revised version, we will include a short calculation in Section 4 confirming that the operator norm is preserved under the relevant mapping, even when the variables do not commute, by using the definition of the operator norm and properties of the joint spectrum. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes direct mathematical proofs of sharp, improved, and refined Bohr inequalities for the class H^∞(D, B(H)) of bounded analytic operator-valued functions, plus multidimensional extensions on complete circular domains Q. These are extensions of Popescu's prior results (distinct authors) rather than reductions to self-citations, fitted parameters, or self-definitional inputs. No equations reduce by construction to the paper's own definitions or prior self-work; the modeling choice of the target space is the intended domain of the theorems, not an unverified assumption. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard definition of bounded analytic operator-valued functions and the geometry of complete circular domains; no free parameters or invented entities are mentioned.

axioms (2)
  • standard math Bounded analytic functions from the unit disk (or complete circular domain) into B(H) satisfy the usual power-series expansion and maximum-modulus properties.
    Invoked when defining the class H^∞(D,B(H)) and when stating the multidimensional analogues.
  • domain assumption The operator norm on B(H) interacts with the scalar Bohr inequality in the expected way.
    Required to lift scalar results to the operator-valued setting.

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Reference graph

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