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The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$

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Pith's one-line read The paper proves the hexablock, a bounded C⁴ domain born from µ-synthesis on upper-triangular 2×2 matrices, is polynomially convex and has distinguished boundary {(a,x₁,x₂,x₃) ∈ C×bE : |a|²+|x₁|² = 1}.

desk verdict Hexablock is a genuinely new C4 domain in the mu-synthesis family with a clean Shilov-boundary theorem, and the visible proofs hold up; the main risk is the computational uniqueness step, not the architecture. read the letter →

arxiv 2506.15149 v2 pith:XGYUYE4O submitted 2025-06-18 math.CV math.FA

Indranil Biswas , Sourav Pal , Nitin Tomar This is my paper · ORCID
classification math.CVmath.FA MSC 30C8032F4593B3693B50
keywords HexablockSymmetrizedbidiscTetrablockPentablockAutomorphismDistinguishedboundarySchwarzlemmaRationalH-innerfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a bounded domain in four complex dimensions, the hexablock H, and argues it is the right finite-dimensional representative of the µ-synthesis problem for 2×2 upper-triangular matrices — the last subspace case in M₂(C), after the symmetrized bidisc, the tetrablock, and the pentablock. The two most naively defined companions, Hμ (image of the structured-singular-value unit ball) and H_N (image of the operator-norm unit ball), turn out to be connected but non-open subsets of C⁴, and a large part of the paper's work is showing that H = int(Hμ) = int(polynomial hull of H_N): the hexablock is the open, bounded, tractable object carrying the same interpolation data. The main payoff is a complete description of the distinguished (Shilov) boundary, bH = {(a,x₁,x₂,x₃) ∈ C × bE : |a|²+|x₁|² = 1}, where E is the tetrablock, along with an explicit subgroup of automorphisms, a full characterization of rational H-inner functions, a Schwarz lemma, and the theorem that the symmetrized bidisc, tetrablock, pentablock, and biball are all analytic retracts of H. A sympathetic reader would care because this gives the theory of the three classical domains a single ambient domain with a transparent boundary, through which results can be transferred and the fourth µ-synthesis case studied concretely.

What carries the argument

The load-bearing object is the two-parameter family of fractional linear maps ψ_{z₁,z₂}(a,x₁,x₂,x₃) = a√((1−|z₁|²)(1−|z₂|²))/(1−x₁z₁−x₂z₂+x₃z₁z₂) on C × E, which generalizes the one-parameter family ψ_z behind the pentablock; the hexablock is simply the set where all |ψ_{z₁,z₂}| < 1 over the tetrablock. Proposition 3.1 shows that for each tetrablock point x the modulus |κ(z₁,z₂,x)| has a unique interior maximizer (z₁(x),z₂(x)) over D², obtained by solving a quadratic whose discriminant factors as a product of four terms (1−|β₁|±|β₂|)(1±|β₁|−|β₂|), and the rejected root is excluded precisely by the tetrablock inequality |β₁|+|β₂| < 1. This yields the continuous function K*(x₁,x₂,x₃), in terms of which membership in H, Hμ, and H_N reduces to |a|K*(x₁,x₂,x₃) < 1, ≤ 1, or the explicit two-sided bounds of Lemma 5.4 respectively. The norm characterization for upper-triangular matrices (Lemma 4.1: ‖X‖ < 1 iff |z₁|,|z₂| < 1 and |w| < √((1−|z₁|²)(1−|z₂|²))) converts operator-norm data into the same shape, and everything downstream — the automorphism group, the boundary computation, the rational inner functions — rests on this reduction of the supremum to a closed form.

What would settle it

Take a generic tetrablock point such as x = (0.3, 0.2, 0.05) in E ∩ R³, compute the two roots of the quadratic (3.7), and check numerically whether the discarded '−' root also lies in D² and gives the same value of |κ| as the '+' root; a second interior critical point at equal or greater height would break the uniqueness on which K* rests. The cheapest decisive test is to numerically maximize |κ(z₁,z₂,x)| over D² for a spread of x in E ∩ R³ and compare with the closed-form value |κ(z₁(x),z₂(x),x)| from Corollary 3.2; any mismatch refutes Proposition 3.1 and hence Theorem 8.21.

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Extended reading notes

Core claim

The paper's central claim is that the hexablock H, defined by sup_{z₁,z₂ ∈ D} |a|√((1−|z₁|²)(1−|z₂|²))/(1−x₁z₁−x₂z₂+x₃z₁z₂) < 1 over the tetrablock E, is the distinguished domain associated with µ-synthesis on the subspace of upper-triangular matrices in M₂(C). Its proof strategy is to study not H directly but its two raw siblings, the µ-hexablock Hμ = {π(A) : μ_hexa(A) < 1} and the normed hexablock H_N = {π(A) : ‖A‖ < 1}, where π(A) = (a₂₁,a₁₁,a₂₂,det A); unlike the older domains, Hμ and H_N are distinct and neither is open, yet the paper proves H = Hμ ∪ ({0}×E), that the closure of H equals the closure of Hμ and equals the polynomial convex hull of H_N, and that H = int(Hμ) = int(polynomial hull of H_N). The boundary theorem (Theorem 8.21) is the centre of the paper: a point lies in the distinguished boundary bH exactly when |a|²+|x₁|² = 1 and (x₁,x₂,x₃) ∈ bE, equivalently when it is a peak point, equivalently when it lies in H_N with |x₃| = 1. From this single description the paper derives the automorphism subgroup G(H), the rational H-inner functions, the Schwarz lemma, and the retraction theorems, and in Chapter 14 re-derives existing pentablock results as consequences.

Load-bearing premise

The load-bearing premise is that the supremum of the kernel |κ(z₁,z₂,x)| over the bidisc is attained at a single interior critical point, singled out by solving a quadratic and discarding the other root with a discriminant inequality; if some tetrablock point produced a second interior critical point (or the discarded root ever became the true maximizer), the function K* and the characterizations of Hμ, H_N, and H built on it would have to be revised.

Editorial extensions

If this is right

  • Membership in H, Hμ, and H_N reduces to a single closed-form inequality: (x₁,x₂,x₃) ∈ E with |a|K*(x₁,x₂,x₃) < 1 for H, with ≤ 1 for Hμ, and with the two-sided bounds of Lemma 5.4 for H_N, so the interpolation data of the µ-synthesis problem are encoded in a bounded domain rather than in the norm-unbounded unit ball.
  • The distinguished boundary of H is exactly the set where |a|²+|x₁|² = 1 over the tetrablock's distinguished boundary — equivalently {( −ξ z, w, ξ w, ξ) : |z|²+|w|² = 1, ξ ∈ T} — giving a complete Shilov boundary, proving H is polynomially convex, and showing H_N is not polynomially convex.
  • Every automorphism of the tetrablock lifts to an automorphism of H, and these lifts together with their flip variants form a subgroup G(H) that preserves Hμ, H_N, the peak-point set H_p, and the three boundary strata ∂₀H, ∂₁H, ∂₂H (Theorems 7.4–7.12 and 8.2).
  • Rational H-inner functions are fully described by rational E-inner functions in the last three coordinates plus an explicit boundedness condition on the first coordinate, and this description yields a Schwarz lemma for two-point interpolation into H (Chapters 12–13).
  • The symmetrized bidisc, the tetrablock, the pentablock, and the biball are all analytic retracts of H, and the paper reproves several known pentablock results by projecting through H (Chapters 10 and 14).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The boundary formula bH = {(a,x₁,x₂,x₃) : |a|²+|x₁|² = 1, (x₁,x₂,x₃) ∈ bE} suggests H is the natural ambient domain for transferring function-theoretic rigidity: any invariant of the tetrablock fibre could be extended to H by averaging over the circle |a|²+|x₁|² = 1, an avenue the paper does not pursue.
  • Because the non-Levi-flat part of ∂E is dense (Theorem 9.10), the methods used to determine the pentablock automorphism group, which rely on Levi-flat boundary strata of the symmetrized bidisc, will not transfer directly; a complete description of Aut(H) will likely require new boundary invariants beyond the Levi form.
  • The strict containments H_N ⊊ Hμ ⊊ H and the fact that Hμ \ H_N contains an open set mark the first instance in the M₂(C) family where the µ-unit ball and the norm unit ball push forward to genuinely different sets; this suggests that for larger upper-triangular subspaces of M_n(C) the gap between the two images will grow, and the 'domain of the µ-synthesis problem' should generally be read as the
  • Proposition 3.1's uniqueness claim is directly checkable by numerical maximization of |κ| over D², so the entire tower of results could be stress-tested independently of the paper's analytic proof; a cheap verification along the real slice E ∩ R³ would either confirm or destabilize the K* formalism.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper introduces the hexablock H, a bounded domain in C^4 defined by a supremum condition over D^2 involving the fractional linear maps ψ_{z1,z2}(a,x1,x2,x3)=a√((1-|z1|^2)(1-|z2|^2))/(1-x1 z1 - x2 z2 + x3 z1 z2), with (x1,x2,x3) ranging over the tetrablock E. It also defines two companion sets, the μ-hexablock H_μ and the normed hexablock H_N, as images of the structured-singular-value unit ball and the operator-norm unit ball of M_2(C) for the upper triangular subspace under the coordinate map π(A)=(a21,a11,a22,det A). The central structural claims are that H is a domain, that H = int(H_μ) and H = H_μ ∪ ({0}×E), and that the closure of H is the polynomial hull of H_N. The main geometric result, Theorem 8.21, identifies the distinguished boundary bH with H_p = {(a,x1,x2,x3) : |a|^2+|x1|^2=1, (x1,x2,x3)∈bE}, and gives eight equivalent characterizations involving unitary matrices, peak points, and boundary points of H_N. The paper also constructs a large subgroup G(H) of Aut(H), proves that any automorphism of H extends past the closure, studies rational H-inner functions and a Schwarz lemma, and shows that B_2, G_2, E and P are analytic retracts of H. The later chapters apply the theory to recover results for the pentablock.

Significance. If the results stand, the paper is a substantive contribution to the μ-synthesis family of domains: it produces a new bounded domain in C^4 with a complete Shilov-boundary description, establishes polynomial convexity and linear convexity, and connects the hexablock with the symmetrized bidisc, tetrablock and pentablock through analytic retracts. The argument is concrete and parameter-free: no fitted constants enter, and the chain from Proposition 3.1, through Theorems 6.1, 6.3, 8.21 and 8.23, is internally coherent rather than imposed. I specifically checked the least machine-verified point, the uniqueness of the maximizer in Proposition 3.1; the discriminant factorization and the rejection of the second root under |β1|+|β2|<1 are valid, and the limiting cases β1=0 or β2=0 are handled consistently. The cited facts about E, G_2 and P are prior results, so I found no circularity. The main caveat is that the copy of the manuscript supplied to me cuts off in Chapter 10, so the pentablock applications in Chapters 11-14 were not independently audited; the central claims of Chapters 1-10 do not depend on those applications.

minor comments (6)
  1. [Proposition 6.9] The set H(r) is displayed as the preimage of the open set H under the anisotropic scaling map, which is an open set, yet the proof then asserts that H(r) is compact and treats it as a compact polynomially convex set. The intended meaning is evidently that an overline on H or H(r) has been lost; please redefine H(r) as a compact exhaustion of H explicitly, for instance as the closure of the scaled set, and adjust the proof accordingly.
  2. [Throughout, especially Theorems 6.3, 5.12, 8.23] The overlines distinguishing H, H_μ, H_N from their closures are missing in several key statements, for example in the statements H = int(H_μ), H_μ = H, and H = int(cH_N). Since the distinction between open hexablock and its closure is load-bearing for these identities, a short notational table or consistent overline placement would remove real ambiguity.
  3. [Proposition 5.7] In the last sentence of the first paragraph of the proof, the text says 'Hence, π(A)∈H_N and so, {π(A) : A∈M_2(C), ||A||≤1}⊆H_μ'; the final inclusion should read '⊆H_N' (or '⊆overline H_N') to follow from the preceding sentence.
  4. [Proposition 8.10] The sentence 'Since q is a peaking function for H' should read 'Since q is a peak point of H'; the surrounding argument is correct but the terminology is nonstandard.
  5. [Proposition 3.1] The derivation of the quadratic equation (3.7) from the critical-point equations (3.6) is summarized as 'some laborious but routine calculations'. Because the uniqueness of the maximizer is the basis for the function K* and hence for all later characterizations of H, including a few displayed intermediate steps or a reference to a companion computation would substantially improve verifiability.
  6. [Theorem 8.21, proof of (7)=>(1)] The step from density of peak points in the Shilov boundary to bH⊆H_p should explicitly use that H_p is compact; as written the conclusion 'bH = P(H)' is stronger than the cited density theorem justifies without the closure argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hexablock construction, its identification with int(H_mu), and the Shilov-boundary theorem are derived from definitions and prior published results without assuming the target claims.

full rationale

The paper's central claims are obtained by explicit derivation rather than by importing the conclusion into the hypotheses. The hexablock H is defined directly by a supremum inequality involving the tetrablock E, separately from the mu-hexablock H_mu and the normed hexablock H_N, which are defined as images of the mu_E-unit ball and the operator-norm unit ball under the coordinate map pi(A)=(a21,a11,a22,det A). Theorem 4.2 derives the characterizations of H_mu from the definition of the structured singular value and the non-vanishing of det(I-AX), with no fitted parameter. Theorem 6.1 and Theorem 6.3 prove H = H_mu union ({0}xE) and H = int(H_mu) by combining the definition of H, the description of int(H_mu) from Proposition 4.8, and continuity of K*, rather than by assuming the equality. The distinguished boundary theorem (Theorem 8.21) is proved through peak-point arguments: H_p is characterized independently via unitary matrices (Proposition 8.3), every point of H_p is shown to be a peak point (Theorem 8.5), and conversely peak points in the three boundary strata are shown to lie in H_p (Propositions 8.15, 8.16, 8.19). The use of the standard theorem that peak points are dense in the Shilov boundary is an external mathematical fact, not a restatement of the paper's conclusion. Theorem 8.23, identifying the polynomial convex hull of H_N with the closure of H, uses the already established facts that H_N is contained in H, that H is polynomially convex, and that bH is contained in H_N; this is a forward implication, not a circular one. Proposition 3.1, the uniqueness of the maximizer of |kappa(z1,z2,x)|, is a computational premise justified in the text by the explicit quadratic and the strict inequality |beta1|+|beta2|<1; it is not equivalent to any target theorem. Citations to prior work on the tetrablock, pentablock, and automorphism groups are published external results used as building blocks, not self-referential justifications of the new claims. No fitted inputs are renamed as predictions, and no definition smuggles the conclusion into an ansatz. The only soft spots, such as the routine extremal computation, are matters of proof scrutiny rather than circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The ledger contains no fitted constants or speculative physical entities. The paper introduces mathematical objects, H, Hµ, and HN, as definitions, not as entities with independent external evidence. The axioms are standard background theorems and prior results from the same research program.

assumptions (4)
  • domain assumption Known characterization theorems for the symmetrized bidisc, tetrablock, and pentablock (Theorems 2.1 through 2.20 in the paper) are correct.
    The hexablock theory is built directly on these prior domains and uses their boundary, polynomial convexity, and linear convexity results without reproving them.
  • domain assumption The structured singular value framework of Doyle and the identification of diagonal, unipotent, and upper triangular structures with E, P, and Hµ are valid.
    The motivation for Hµ and HN relies on the mu-synthesis equivalence between interpolation problems and domains, as established in earlier papers on G2, E, and P.
  • domain assumption Proper holomorphic maps between quasi-balanced domains extend past the closure and preserve the Shilov boundary, as cited from reference [58].
    This external theorem drives the automorphism extension and Shilov boundary preservation arguments in Propositions 7.13, 8.11, and 9.1.
  • standard math Standard results in several complex variables and function theory, including the maximum modulus principle, open mapping theorem, polynomial convexity of compact convex sets, and density of peak points in the Shilov boundary.
    These are used throughout the paper without proof, which is normal for a research-level complex analysis paper.

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Pith. "Pith review of The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$." pith.science (2026). https://pith.science/paper/XGYUYE4O

@misc{pith2026250615149,
  author       = {Pith},
  title        = {Pith review of: The Hexablock: a domain associated with the $\mu$-synthesis in $M_2(\mathbb C)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XGYUYE4O}},
  note         = {Machine review of arXiv:2506.15149}
}
abstract

We introduce a domain named \textit{hexablock} in $\mathbb C^4$ and show that its origin is a special case of $\mu$-synthesis in $M_2(\mathbb C)$, more precisely the $\mu_E$-unit ball with respect to the linear subspace $E$ consisting of $2 \times 2$ upper triangular matrices. The hexablock is denoted by $\mathbb H$ and is defined by \[ \mathbb{H}=\left\{(a, x_1, x_2, x_3) \,\in\, \mathbb{C} \times \mathbb{E}\,\,\big\vert\,\, \sup_{z_1,\, z_2 \,\in\, \mathbb D}\left|\frac{a\sqrt{(1-|z_1|^2)(1-|z_2|^2)}}{1-x_1z_1-x_2z_2+x_3z_1z_2}\right| <1\right\}, \] where $\mathbb{E}$ is the \textit{tetrablock}, another domain in $\mathbb C^3$ associated with a different case of $\mu$-synthesis, and is given by \[ \mathbb{E}=\{(x_1, x_2, x_3) \in \mathbb{C}^3 : 1-x_1z_1-x_2z_2+x_3z_1z_2 \ne 0 \ \text{for all } \, z_1, z_2 \in \overline{\mathbb D}\}. \] We show that two other objects in $\mathbb C^4$ namely, the $\mu$-hexablock $\mathbb H_{\mu}$ and the normed hexablock $\mathbb H_N$ naturally arise in the $\mu_E$-unit ball and the norm unit ball of $M_2(\mathbb C)$, respectively and pave the way to reach the domain $\mathbb H$. A set of independent characterizations for the points in $\mathbb H_{\mu}, \mathbb H_N$ and $\mathbb H$ are obtained. Geometric and function theoretic aspects of $\mathbb H$ are studied and its connections with the popular domains such as symmetrized bidisc $\mathbb G_2$, tetrablock $\mathbb E$ and pentablock $\mathbb P$ are explored.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Operators associated with a domain in $\mathbb C^4$ and applications

    math.FA 2025-07 accept novelty 7.0 of 10

    H-contractions, commuting quadruples with the hexablock as spectral set, are characterized through ball and tetrablock contractions, with Wold decomposition, conditional dilation, and canonical decomposition.

  2. Function theory of the hexablock and applications to the tetrablock and Euclidean biball

    math.CV 2026-08 conditional novelty 6.0 of 10

    The paper proves realization, interpolation, extension, and Toeplitz corona theorems for the hexablock, recovering the tetrablock and biball results as special cases.

  3. Rigidity of proper holomorphic self-mappings of the hexablock

    math.CV 2025-07 conditional novelty 6.0 of 10

    Every proper holomorphic self-map of the hexablock is an automorphism, equal to one of the explicit maps (1.3) or (1.4), establishing G(H) = Aut(H).

Reference graph

Works this paper leans on

94 extracted references · 78 canonical work pages · cited by 3 Pith papers

  1. [1]

    A. A. Abouhajar, Function theory related to H∞ control, PhD Thesis, Newcastle University, 2007

    2007

  2. [2]

    A. A. Abouhajar, M. C. White and N. J. Young, A Schwarz lemma for a domain related to µ- synthesis, J. Geom. Anal., 17 (2007), 717 – 750

    2007

  3. [3]

    Agler, Z

    J. Agler, Z. A. Lykova and N. J. Young, Extremal holomorphic maps and the symmetrized bidisc , Proc. London Math. Soc., 106 (2013), 781 – 818

    2013

  4. [4]

    Agler, Z

    J. Agler, Z. A. Lykova and N. J. Young, The complex geometry of a domain related to µ-synthesis, J. Math. Anal. Appl., 422 (2015), 508 – 543

    2015

  5. [5]

    Agler, Z

    J. Agler, Z. A. Lykova and N. J. Young, Algebraic and geometric aspects of rational Γ-inner func- tions, Adv. Math., 328 (2018), 133 – 159

    2018

  6. [6]

    Agler, Z

    J. Agler, Z. A. Lykova and N. J. Young, Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc, Mem. Amer. Math. Soc., 258 (2019), no. 1242

    2019

  7. [7]

    Agler and N

    J. Agler and N. J. Young, A commutant lifting theorem for a domain in C2 and spectral interpola- tion, J. Funct. Anal., 161 (1999), 452 – 477

    1999

  8. [8]

    Agler and N

    J. Agler and N. J. Young, The two-point spectral Nevanlinna-Pick problem, Integral Equations Op- erator Theory, 37 (2000), 375 – 385

    2000

Show all 94 references
  1. [9]

    Agler and N

    J. Agler and N. J. Young, A Schwarz lemma for the symmetrised bidisc , Bull. London Math. Soc., 33 (2001), 175 – 186

    2001

  2. [10]

    Agler and N

    J. Agler and N. J. Young, A model theory for Γ-contractions, J. Operator Theory, 49 (2003), 45 – 60

    2003

  3. [11]

    Agler and N

    J. Agler and N. J. Young, The hyperbolic geometry of the symmetrized bidisc , J. Geom. Anal., 14 (2004), 375 – 403

    2004

  4. [12]

    Agler and N

    J. Agler and N. J. Young, The two-by-two spectral Nevanlinna-Pick problem, Trans. Amer. Math. Soc., 356 (2004), 573 – 585

    2004

  5. [13]

    Agler and N

    J. Agler and N. J. Young, The magic functions and automorphisms of a domain , Complex Anal. Oper. Theory, 2 (2008), 383 – 404

    2008

  6. [14]

    O. M. O. Alsalhi and Z. A. Lykova, Rational tetra-inner functions and the special variety of the tetrablock, J. Math. Anal. Appl., 506 (2022), Paper No. 125534, 52 pp

    2022

  7. [15]

    N. M. Alshehri and Z. A. Lykova, A Schwarz lemma for the pentablock, J. Geom. Anal., 33 (2023), Paper No. 65, 44 pp

    2023

  8. [16]

    J. A. Ball, I. Gohberg and L. Rodman, Interpolation of rational matrix functions, OT45, Birkhauser Verlag, 1990

    1990

  9. [17]

    J. A. Ball and H. Sau, Rational dilation of tetrablock contractions revisited , J. Funct. Anal., 278 (2020), no. 1, 108275, 14 pp

    2020

  10. [18]

    M. S. Baouendi, P. Ebenfelt and L. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, V ol. 47, Princeton University Press, Princeton, NJ, 1999

    1999

  11. [19]

    Bharali, Some new observations on interpolation in the spectral unit ball , Integral Equations Operator Theory, 59 (2007), 329 – 343

    G. Bharali, Some new observations on interpolation in the spectral unit ball , Integral Equations Operator Theory, 59 (2007), 329 – 343. [132] Hexablock 133

    2007

  12. [20]

    Bharali, A family of domains associated with µ-synthesis, Integral Equations Operator Theory, 82 (2015), 267 – 285

    G. Bharali, A family of domains associated with µ-synthesis, Integral Equations Operator Theory, 82 (2015), 267 – 285

    2015

  13. [21]

    Bhattacharyya, S

    T. Bhattacharyya, S. Pal and S. Shyam Roy, Dilations of Γ-contractions by solving operator equa- tions, Adv. Math., 230 (2012), 577 – 606

    2012

  14. [22]

    Bhattacharyya, The tetrablock as a spectral set, Indiana Univ

    T. Bhattacharyya, The tetrablock as a spectral set, Indiana Univ. Math. J., 63 (2014), 1601 – 1629

    2014

  15. [23]

    Bhattacharyya and S

    T. Bhattacharyya and S. Pal, A functional model for pure Γ-contractions, J. Operator Theory, 71 (2014), 327 – 339

    2014

  16. [24]

    Bhattacharyya and H

    T. Bhattacharyya and H. Sau, Holomorphic functions on the symmetrized bidisk-realization, inter- polation and extension, J. Funct. Anal., 274 (2018), 504 – 524

    2018

  17. [25]

    Bisai and S

    B. Bisai and S. Pal, Structure theorems for operators associated with two domains related to µ- synthesis, Bull. Sci. Math., 159 (2020), Paper No. 102822, 30 pp

    2020

  18. [26]

    Bisai and S

    B. Bisai and S. Pal, A model theory for operators associated with a domain related to µ-synthesis, Collect. Math., 74 (2023), 173 – 198

    2023

  19. [27]

    Blaschke, Ein Erweiterung des Satzes von Vitali uber Folgen analytischer Funktionen, Berichte Math.-Phys

    W. Blaschke, Ein Erweiterung des Satzes von Vitali uber Folgen analytischer Funktionen, Berichte Math.-Phys. Kl., Sachs. Gesell. der Wiss. Leipzig, 67 (1915), 194 – 200

    1915

  20. [28]

    Browder, Introduction to Function Algebras, W

    A. Browder, Introduction to Function Algebras, W. A. Benjamin Inc., New York, 1969

    1969

  21. [29]

    D. C. Brown, A rich structure related to the construction of analytic matrix functions, PhD Thesis, Newcastle University, 2016

    2016

  22. [30]

    D. C. Brown, Z. A. Lykova and N. J. Young, A rich structure related to the construction of analytic matrix functions, J. Funct. Anal., 272 (2017), 1704 – 1754

    2017

  23. [31]

    Carath ´eodory, ¨Uber den variabilit ¨atsbereich der fourier’schen konstanten von positiven har- monischen funktionen, Rend

    C. Carath ´eodory, ¨Uber den variabilit ¨atsbereich der fourier’schen konstanten von positiven har- monischen funktionen, Rend. Circ. Matem. Palermo, 32 (1911), 193 – 217

    1911

  24. [32]

    J. L. Clerc, Geometry of the Shilov boundary of a bounded domain , J. Geom. Symmetry Phys., 13 (2009), 25 – 74

    2009

  25. [33]

    Costara, The symmetrized bidisc and Lempert’s theorem , Bull

    C. Costara, The symmetrized bidisc and Lempert’s theorem , Bull. London Math. Soc., 36 (2004), 656 – 662

    2004

  26. [34]

    Costara, The 2× 2 spectral Nevanlinna-Pick problem, J

    C. Costara, The 2× 2 spectral Nevanlinna-Pick problem, J. London Math. Soc., 71 (2005), 684 – 702

    2005

  27. [35]

    Costara, On the spectral Nevanlinna-Pick problem, Studia Math., 170 (2005), 23 – 55

    C. Costara, On the spectral Nevanlinna-Pick problem, Studia Math., 170 (2005), 23 – 55

    2005

  28. [36]

    Doyle, Analysis of feedback systems with structured uncertainties , IEE Proc

    J. Doyle, Analysis of feedback systems with structured uncertainties , IEE Proc. Control Theory Appl., 129 (1982), 242 – 250

    1982

  29. [37]

    J. C. Doyle and G. Stein, Multivariable feedback design: concepts for a classical/modern synthesis, IEEE Transactions on Automatic Control, 26 (1981), 4 – 16

    1981

  30. [38]

    P. L. Duren, Theory of H p spaces, Academic Press, New York, (1970)

    1970

  31. [39]

    Edigarian, A family of Lempert domains, J

    A. Edigarian, A family of Lempert domains, J. Geom. Anal., 34 (2024), no. 6, Paper no. 176, 9 pp

    2024

  32. [40]

    Edigarian, L

    A. Edigarian, L. Kosi ´nski and W. Zwonek,The Lempert theorem and the tetrablock, J. Geom. Anal., 23 (2013), 1818 – 1831

    2013

  33. [41]

    Edigarian and W

    A. Edigarian and W. Zwonek, Geometry of the symmetrized polydisc , Arch. Math. (Basel), 84 (2005), 364 – 374

    2005

  34. [42]

    Edigarian and W

    A. Edigarian and W. Zwonek, Schwarz lemma for the tetrablock, Bull. Lond. Math. Soc., 41 (2009), 506 – 514

    2009

  35. [43]

    Foias and A

    C. Foias and A. E. Frazho, The commutant lifting approach to interpolation problem , Birkh¨auser, Berlin, 1996

    1996

  36. [44]

    G. B. Folland, Introduction to partial differential equations , V ol. 102, Princeton university press, 2020

    2020

  37. [45]

    B. A. Francis, A course in H∞ Control Theory, Lecture Notes in Control and Information Sciences, V ol. 88, Springer, Berlin (1987). 134 I. Biswas, S. Pal and N. Tomar

    1987

  38. [46]

    Fritzsche and H

    K. Fritzsche and H. Grauert, From holomorphic functions to complex manifolds, Graduate Text in Mathematics, Springer 2002

    2002

  39. [47]

    B. A. Fuks, Special chapters in the theory of analytic functions of several complex variables, Trans- lations of Mathematical Monographs 14 (American Mathematical Society, Providence RI, 1965)

    1965

  40. [48]

    Gelfand, D

    I. Gelfand, D. Raikov and G. Shilov, Commutative normed rings, Chelsea (1964)

    1964

  41. [49]

    Ghosh and W

    G. Ghosh and W. Zwonek, Holomorphic retracts in the Lie ball and the tetrablock, J. Geom. Anal., 35 (2025), no. 4, Paper No. 118, 17 pp

    2025

  42. [50]

    W. H. Gottschalk, Intersection and Closure, Proc. Amer. Math. Soc., 4 (3) (1953), 470 – 473

    1953

  43. [51]

    Grimm, Rational Approximation of Transfer Functions in the Hyperion Software , Research Re- port RR-4002, INRIA (2000)

    J. Grimm, Rational Approximation of Transfer Functions in the Hyperion Software , Research Re- port RR-4002, INRIA (2000)

    2000

  44. [52]

    L. A. Harris, Bounded symmetric homogeneous domains in infinite-dimensional spaces , Proceed- ings on Infinite Dimensional Holomorphy, Springer Lecture Notes in Mathematics, 364 (1973), 13 – 40

    1973

  45. [53]

    L. A. Harris, Analytic invariants and the Schwarz-Pick inequality, Israel J. Math., 34 (1979), 177 – 197

    1979

  46. [54]

    T. G. Honary, The density of peak points in the Shilov Boundary of a Banach fuction algebra, Proc. Amer. Math. Soc., V ol. 103, 1988

    1988

  47. [55]

    Jarnicki and P

    M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, 2nd edition, Walter de Gruyter, Berlin-Newyork, 2013

    2013

  48. [56]

    Jindal and P

    A. Jindal and P. Kumar, Rational penta-inner functions and the distinguished boundary of the pentablock, Complex Anal. Oper. Theory, 16 (2022), Paper No. 120, 12 pp

    2022

  49. [57]

    Jindal and P

    A. Jindal and P. Kumar, Operator theory on the pentablock, J. Math. Anal. Appl., 540 (2024), no. 1, Paper No. 128589, 17 pp

    2024

  50. [58]

    Kosi ´nski, Geometry of quasi-circular domains and applications to tetrablock, Proc

    L. Kosi ´nski, Geometry of quasi-circular domains and applications to tetrablock, Proc. Amer. Math. Soc., 139 (2011), 559 – 569

    2011

  51. [59]

    Kosi ´nski, The group of automorphisms of the pentablock, Complex Anal

    L. Kosi ´nski, The group of automorphisms of the pentablock, Complex Anal. Oper. Theory, 6 (2015), 1349 – 1359

    2015

  52. [60]

    Kosi ´nski and W

    L. Kosi ´nski and W. Zwonek, Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock, J. Geom. Anal., 26 (2016), 1863 – 1890

    2016

  53. [61]

    S. G. Krantz, Function Theory of Several Complex Variables , John Wiley and Sons, New York, 1982

    1982

  54. [62]

    Lempert, La m ´etrique de Kobayashi et la repr ´esentation des domaines sur la boule (French

    L. Lempert, La m ´etrique de Kobayashi et la repr ´esentation des domaines sur la boule (French. English summary), Bull. Soc. Math. France, 109 (1981), 427 – 474

    1981

  55. [63]

    Mackey and P

    M. Mackey and P. Mellon, The Bergmann-Shilov boundary of a bounded symmetric domain, Math. Proc. R. Ir. Acad., 121 (2021), 33 – 49

    2021

  56. [64]

    Mas-Colell, M

    A. Mas-Colell, M. D. Whinston and J. R. Green, Microeconomic theory, New York: Oxford univer- sity press (1995)

    1995

  57. [65]

    Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann

    N. Nikolov, The symmetrized polydisc cannot be exhausted by domains biholomorphic to convex domains, Ann. Pol. Math., 88 (2006), 279 – 283

    2006

  58. [66]

    Nikolov, P

    N. Nikolov, P. Pflug and P. J. Thomas, Spectral Nevanlinna-Pick and Carath´eodory-Fej´er problems for n≤ 3, Indiana Univ. Math. J., 60 (2011), 883 – 893

    2011

  59. [67]

    Nikolov, P

    N. Nikolov, P. Pflug, P. J. Thomas and W. Zwonek, Estimates of the Caratheodory metric on the symmetrized polydisc, J. Math. Anal. Appl., 341 (2008), 140 – 148

    2008

  60. [68]

    Nikolov, P

    N. Nikolov, P. Pflug and W. Zwonek, The Lempert function of the symmetrized polydisc in higher dimensions is not a distance, Proc. Amer. Math. Soc., 135 (2007), 2921 – 2928

    2007

  61. [69]

    Nikolov, P

    N. Nikolov, P. Pflug and W. Zwonek, An example of a bounded C-convex domain which is not biholomorphic to a convex domain, Math. Scand., 102 (2008), 149 – 155. Hexablock 135

    2008

  62. [70]

    D. J. Ogle, Operator and function theory of the symmetrized polydisc, PhD Thesis, Newcastle Uni- versity, 1999

    1999

  63. [71]

    Pal and O

    S. Pal and O. M. Shalit, Spectral sets and distinguished varieties in the symmetrized bidisc, J. Funct. Anal., 266 (2014), 5779 – 5800

    2014

  64. [72]

    Pal, From Stinespring dilation to Sz.-Nagy dilation on the symmetrized bidisc and operator mod- els, New York J

    S. Pal, From Stinespring dilation to Sz.-Nagy dilation on the symmetrized bidisc and operator mod- els, New York J. Math., 20 (2014), 645 – 664

    2014

  65. [73]

    Pal, The failure of rational dilation on the tetrablock, J

    S. Pal, The failure of rational dilation on the tetrablock, J. Funct. Anal., 269 (2015), 1903 – 1924

    2015

  66. [74]

    Pal, Subvarieties of the tetrablock and von Neumann’s inequality , Indiana Univ

    S. Pal, Subvarieties of the tetrablock and von Neumann’s inequality , Indiana Univ. Math. J., 65 (2016), 2051 – 2079

    2016

  67. [75]

    Pal, Canonical decomposition of a tetrablock contraction and operator model , J

    S. Pal, Canonical decomposition of a tetrablock contraction and operator model , J. Math. Anal. Appl., 438 (2016), 274 – 284

    2016

  68. [76]

    Pal and S

    S. Pal and S. Roy, A generalized Schwarz lemma for two domains related to µ-synthesis, Complex Manifolds, 5 (2018), 1 – 8

    2018

  69. [77]

    Pal, Canonical decomposition of operators associated with the symmetrized polydisc , Complex Anal

    S. Pal, Canonical decomposition of operators associated with the symmetrized polydisc , Complex Anal. Oper. Theory, 12 (2018), 931 – 943

    2018

  70. [78]

    Pal and S

    S. Pal and S. Roy, A Schwarz lemma for two families of domains and complex geometry , Complex Var. Elliptic Equ., 66 (2021), 756 – 782

    2021

  71. [79]

    Pal and N

    S. Pal and N. Tomar, Operators associated with the pentablock and their relations with biball and symmetrized bidisc, https://arxiv.org/abs/2309.15080v2

    arXiv

  72. [80]

    Pflug and W

    P. Pflug and W. Zwonek, Description of all complex geodesics in the symmetrised bidisc , Bull. London Math. Soc., 37 (2005), 575 – 584

    2005

  73. [81]

    Riesz and B

    F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York (1990)

    1990

  74. [82]

    J. C. Robinson, An introduction to functional analysis, Cambridge University Press, 2020

    2020

  75. [83]

    R. T. Rockafellar, Convex analysis, Princeton University Press, Princeton, NJ, 1997

    1997

  76. [84]

    Rudin, Functional analysis, Second edition, Internat

    W. Rudin, Functional analysis, Second edition, Internat. Ser. Pure Appl. Math. McGraw-Hill, Inc., New York, 1991

    1991

  77. [85]

    G. Su, Z. Tu and L. Wang, Rigidity of proper holomorphic self-mappings of the pentablock, J. Math. Anal. Appl., 424 (2015), 460 – 469

    2015

  78. [86]

    Su, Geometric properties of the pentablock, Complex Anal

    G. Su, Geometric properties of the pentablock, Complex Anal. Oper. Theory, 14 (2020), Paper no. 44, 14 pp

    2020

  79. [87]

    Sz.-Nagy, C

    B. Sz.-Nagy, C. Foias, L. Kerchy and H. Bercovici, Harmonic analysis of operators on Hilbert space, Universitext Springer, New York, 2010

    2010

  80. [88]

    P. J. Thomas, N. V . Trao and W. Zwonek, Green functions of the spectral ball and symmetrized polydisk, J. Math. Anal. Appl., 377 (2011), 624 – 630

    2011

  81. [89]

    Tomar, A new characterization for the distinguished boundaries of a few domains related to µ-synthesis, Proc

    N. Tomar, A new characterization for the distinguished boundaries of a few domains related to µ-synthesis, Proc. Indian Acad. Sci. (Math. Sci.), 2024

    2024

  82. [90]

    V oigt, A Course on Topological Vector Spaces, V ol

    J. V oigt, A Course on Topological Vector Spaces, V ol. 20020, Cham: Birkh¨auser, 2020

    2020

  83. [91]

    N. J. Young, An Introduction to Hilbert Space, Cambridge University Press, Cambridge, (1988)

    1988

  84. [92]

    N. J. Young, The automorphism group of the tetrablock, J. Lond. Math. Soc., 77 (2008), 757 – 770

    2008

  85. [93]

    Zapałowski, Geometric properties of domains related to µ-synthesis, J

    P. Zapałowski, Geometric properties of domains related to µ-synthesis, J. Math. Anal. Appl., 430 (2015), 126 – 143

    2015

  86. [94]

    Zwonek, Geometric properties of the tetrablock, Arch

    W. Zwonek, Geometric properties of the tetrablock, Arch. Math. (Basel), 100 (2013), 159 – 165

    2013

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