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Function theoretic aspects of the symmetrized polydisc and generalization

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Pith's one-line read This paper proves that the Schur-Agler class on the symmetrized polydisc is exactly the functions with unitary-colligation form built from the rational function J(s), and derives interpolation, corona, and extension theorems from that.

desk verdict Useful extension of Agler-kernel machinery to G_d, but the manuscript has a systematic conjugation bug and Section 4 is underproved; worth reviewing after fixes. read the letter →

arxiv 2608.00827 v1 pith:XB744PLN submitted 2026-08-01 math.CV math.FAmath.OA

Sourav Pal , Nitin Tomar This is my paper
classification math.CVmath.FAmath.OA MSC 32A7047A1347A5746E2247B32
keywords Schur-AglerclasssymmetrizedpolydiscrealizationtheoremNevanlinna-PickinterpolationToeplitzcoronanorm-preservingextensionadmissiblekernelsgeneralizeddomains
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

This paper aims to establish a function-theoretic foundation for the symmetrized polydisc G_d, the d-dimensional domain obtained from the unit polydisc by taking elementary symmetric polynomials of its coordinates. The main target is a realization theorem: a holomorphic function f on G_d with values in the bounded operators between two Hilbert spaces belongs to the Schur-Agler class—defined by the requirement that the norm of f(S) is at most 1 for every commuting operator tuple S whose joint spectrum sits in G_d and whose associated rational functions are contractions—if and only if f can be written as A + Bρ(J(s))(I − Dρ(J(s)))^{-1}C, with one rational function J controlling the formula. The same equivalence reaches kernel positivity and completely positive maps, and from it the paper derives a Nevanlinna-Pick interpolation theorem, a Toeplitz corona theorem, and a norm-preserving extension theorem for G_d. These are the higher-dimensional analogues of results known for the symmetrized bidisc, and together they make G_d a domain where interpolation, corona, and extension are governed by one realization principle. The final section extends the package to a family of generalized symmetrized domains Θ_d, with the understanding that the proofs are essentially the same.

What carries the argument

The central object is the family of rational functions Φ_α(s) = J(s)(α), where J(s) = Q(s)/R(s) and Q, R are the polynomials Q(s)(α) = d(−1)^d p α^{d−1} + ... + (−s_1) and R(s)(α) = d − (d−1)s_1 α + ... + (−1)^{d−1} s_{d−1} α^{d−1}; G_d is exactly the set of points with sup_α |Φ_α(s)| < 1. The mechanism that carries the argument is the standard kernel-to-realization route for Schur-Agler classes: admissible kernels are defined by the positivity of (1 − Φ_α(s)conjugate(Φ_α(t)))k(s,t), and the proof shows the Schur-Agler norm condition forces a completely positive map ξ, which in turn yields the unitary-colligation formula. In that formula ρ(J(s)) is the image of the single rational function J

What would settle it

With d=3, m=6, p=2, take a point θ in Θ_d and check whether (r θ_1, r^2 θ_2, r^{3/2} θ_3) remains in Θ_d for every r in (0,1); equivalently, check whether the associated rational functions stay contractions under this scaling. If some point or operator tuple in M_{Θ_d} violates this scaling, the proof route used for G_d cannot transfer verbatim to Θ_d, and the Section 4 theorems need a separate argument.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is an equivalence of four descriptions of a holomorphic vector-valued function f on G_d: being in the Schur-Agler class; making (I − f(s)f(t)^*) ⊗ k(s,t) positive semidefinite for every admissible kernel k; having the defect I − f(s)f(t)^* equal to ξ(s,t)(1 − J(s)conjugate(J(t))) for a completely positive map ξ; and admitting a transfer-function realization f(s) = A + Bρ(J(s))(I − Dρ(J(s)))^{-1}C with a unitary block operator V and a unital *-representation ρ of the continuous functions on the disc. The function J(s) = Q(s)/R(s) is built from the coordinate polynomials Q and R, and a point s lies in G_d exactly when |J(s)(α)| < 1 for every α in the dis

Load-bearing premise

The transfer from G_d to Θ_d rests on an unstated assumption: the operator class M_{Θ_d} admits the same radial scaling used in the G_d proofs, with the last coordinate scaled by r^{d/p} rather than r^d; if that scaling fails for some p≠1, the 'proofs essentially identical' claim does not establish the Section 4 theorems.

Editorial extensions

If this is right

  • Every function in the Schur-Agler class on G_d has a unitary-colligation representation, and every such representation lies in the class, so the two descriptions coincide.
  • Nevanlinna-Pick interpolation on G_d has a necessary and sufficient criterion: an interpolant exists in the Schur-Agler class exactly when the data satisfy the complete-positivity defect equation, equivalently positivity against every admissible kernel.
  • A Toeplitz corona equation ζ12 ζ31 = ζ32 has a Schur-Agler solution exactly when the operator-valued kernel built from ζ12 and ζ32 is positive against all admissible kernels.
  • Norm-preserving extension: a function f on a subset U0 extends to a function whose scalar multiple lies in the Schur-Agler class iff every operator tuple subordinate to U0 satisfies the norm bound ∥f(S)∥ ≤ ∥f∥∞,U0.
  • The same four equivalences—realization, kernel positivity, complete positivity, and unitary colligation—are stated for the generalized symmetrized domains Θ_d, with J replaced by the analogue J_*.

Reading between the lines

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

  • The realization formula suggests that the natural dilation/model theory for tuples in M_{G_d} should be derivable from the unitary V, though the paper does not state a dilation theorem.
  • If the Section 4 scaling gap is real, the Θ_d theorems might still be true but would need a proof that replaces quasi-balanced scaling with a homogeneity-aware argument; the first place to look is the implication (1) => (2) of the realization theorem on Θ_d.
  • The interpolation criterion in Theorem 3.1 is a finite system of operator inequalities, so it can in principle be used numerically to test interpolation data on G_d for small d; the paper does not explore algorithms.
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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

2 major / 4 minor

Summary. The paper introduces a Schur-Agler class for the symmetrized polydisc G_d, defined as holomorphic operator-valued functions f such that ||f(S)|| ≤ 1 for every commuting d-tuple S whose joint spectrum lies in G_d and for which ||Φ_α(S)|| < 1 for all α in the unit disc. The central result (Theorem 2.8) asserts the equivalence of this class with an admissible-kernel positivity condition, a completely positive map factorization of 1 − f(s)f(t)^*, and a unitary-colligation realization built from a representation of C(∂D). From this equivalence the paper derives an interpolation theorem, a Toeplitz corona theorem, and a norm-preserving extension theorem for G_d (Theorems 3.1–3.3), and then states analogous results for the generalized domains Θ_d (Theorems 4.2–4.5), asserting that the proofs are essentially identical.

Significance. If the main theorem is correct, this is a meaningful extension of the Agler realization method to the symmetrized polydisc in arbitrary dimension, unifying and generalizing prior work on G_2, the tetrablock, and the pentablock. The applications to interpolation, corona, and extension are natural and potentially useful. The paper is honest about its debts: Theorem 2.1 is explicitly credited to Costara's characterization, and the overall architecture follows the standard Agler/McCullough framework. Several parts, especially the approximation argument in (4)=>(1) of Theorem 2.8, are carried out in detail. However, the manuscript currently contains a systematic conjugation error in the definition of admissibility and an unproved—but load-bearing—scaling assertion for the operator class M_G_d and its Θ_d analogue. Both issues are fixable, but they must be addressed before the central claims can be accepted.

major comments (2)
  1. [§2, Definition 2.3; Theorem 2.8(2)–(3); also Theorems 3.1, 3.2, §4] The admissible kernel condition is written as (s,t) ↦ (1 − Φ_α(s)Φ_α(t))k(s,t) being positive semidefinite, and the same un-conjugated expression 1−J(s)J(t) appears in Lemma 2.5, Theorem 2.8(3), Theorem 3.1(3), Theorem 3.2(3), and the Section 4 analogues. For a positive semidefinite kernel this factor must be 1 − Φ_α(s)overline{Φ_α(t)} (equivalently 1−J(s)overline{J(t)}); as written, the kernel is not even Hermitian, and the pullback of the Szegő kernel is misidentified. This is not a cosmetic issue: the positivity condition is the bridge between the operator class M_G_d and the realization formula. In the proof of Theorem 2.8(3)=>(4), after correcting the conjugation the displayed algebra should involve ρ(J(s))ρ(J(t)), not ρ(J(s))ρ(J(t))^*. The authors should correct this consistently throughout the paper and re-verify the isometry step.
  2. [§2, property (3) of M_G_d; Used in Theorems 2.8, 3.2, 3.3; §4] The assertion that (rS_1, ..., r^{d-1}S_{d-1}, r^dP) ∈ M_G_d for 0<r<1 when S ∈ M_G_d is stated as 'evident' from the scalar quasi-balancedness of G_d. Scalar quasi-balancedness does not by itself imply the required operator inequality ||Φ_α(rS_1,...,r^dP)|| < 1. A proof is needed, for example via the identity Φ_α(r·s) = rΦ_{rα}(s), which is not stated. This step is used critically in Theorem 2.8(1)=>(2), Theorem 3.2(1)=>(2), and Theorem 3.3. For Θ_d, no analogue of this scaling is stated or proved at all, yet Theorems 4.2–4.5 are said to follow with 'minor modifications.' The authors should state and prove the scaling lemma for M_G_d and for M_Θ_d explicitly; without it, the transfer to Θ_d is unsupported.
minor comments (4)
  1. [§2, Theorem 2.1 proof] After proving |J(s)(α)|<1 for every α∈D and continuity on the closed disc, the proof concludes that ||J(s)||_{∞,D} = |J(s)(α)| for some α∈D. The maximum on the closed disc can occur on the boundary, and boundary behavior has not been controlled. Please complete this argument or cite Costara's theorem directly for the uniform bound.
  2. [§3, proof of Theorem 3.2] In the subspace H_n(k), the index is written as 1≤ℓ≤m although the finite set F has n points. This is a typo but should be corrected for readability.
  3. [§4, definitions] The definition of Θ_d is printed with an ellipsis that obscures the role of the last coordinate: θ_i = s_i(z_1^m,...,z_d^m) for 1≤i≤d−1 and θ_d = (z_1...z_d)^{m/p}. The displayed formula should be cleaned up. Also in the definition of P_*(θ), the final term appears garbled as θ_p_d; it should be θ_d^p.
  4. [§2, proof of Theorem 2.8(4)=>(1)] The text says there is a 'subsequence' of the net {f_β}; nets have subnets, not subsequences. The argument is standard, but the terminology should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the realization theorem is an independent Agler-type argument; the only flagged issue is an unproved scaling lemma, which is a rigor gap, not a circular reduction.

full rationale

The paper's load-bearing claim is Theorem 2.8, a four-way equivalence between the operator-tuple Schur-Agler class, an admissible-kernel positivity condition, a completely positive map factorization, and a unitary-colligation realization. These conditions are not definitionally identical: the implications (1)=>(2) and (4)=>(1) are proved by RKHS, functional-calculus, and approximation arguments, and the theorem is not a renaming of the definition of SA_Gd. Theorem 2.1 is explicitly presented as a repackaging of Costara's characterization [29], with an acknowledgment that it follows from that prior work. Self-citations [46]-[48] are used only for background on the pentablock or as one of several standard references for a proof that is actually included in the text; they are not load-bearing. The one flagged issue is Section 2, item (3): 'Evidently, the domain G_d is (1,...,d-1,d)-quasi-balanced... Therefore, (rS_1,...,r^{d-1}S_{d-1},r^dP) ∈ M_Gd' — an asserted but unproved operator-norm lemma used in Theorem 2.8(1)=>(2), Theorem 3.2(1)=>(2), and Theorem 3.3, with the Θ_d analogue in Section 4 not stated. This is a rigor/correctness gap, not circularity: the assertion does not reduce a conclusion to an input, and it is repairable by the (unstated) identity Φ_α(r·s)=rΦ_{rα}(s). The derivation chain is otherwise self-contained against standard external results.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical fitting is involved. The paper consumes Costara's and [39]'s characterizations as external inputs, assumes standard functional calculus and kernel-space machinery, and introduces new function and operator classes but no new physical or geometric entities. The main unstated load is the scaling assumption for Θ_d.

assumptions (5)
  • domain assumption Costara's characterization: s∈G_d iff ∥J(s)∥∞,D<1 (Theorem 3.1 in [29])
    Used at the start of Section 2 to define Φ_α and to prove Theorem 2.1; the paper's own proof of Theorem 2.1 is a modification of [29].
  • standard math Spectral mapping and rational functional calculus for commuting tuples
    Used to pass from scalar inequalities |Φ_α(s)|<1 to operator inequalities for tuples in M_{G_d} in Section 2.
  • domain assumption Keshari-Nayak-Pal-Paul characterization of Θ_d: θ∈Θ_d iff sup_{α∈D}|Φ_{*,α}(θ)|<1 (Theorem 3.1 in [39])
    Basis for the entire Section 4; cited but not proved.
  • domain assumption Quasi-balanced scaling r·S∈M_{G_d} for 0<r<1
    Used throughout Sections 2 and 3 to perturb finite-dimensional restrictions into the strict class M_{G_d}. The analogue for Θ_d is unstated.
  • standard math Kurosh's theorem and inverse limit representation for completely positive maps
    Invoked at the end of Lemma 2.5 proof to pass from finite subsets to all of G_d.

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Pith. "Pith review of Function theoretic aspects of the symmetrized polydisc and generalization." pith.science (2026). https://pith.science/paper/XB744PLN

@misc{pith2026260800827,
  author       = {Pith},
  title        = {Pith review of: Function theoretic aspects of the symmetrized polydisc and generalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XB744PLN}},
  note         = {Machine review of arXiv:2608.00827}
}
abstract

We introduce Schur-Agler type class for the symmetrized polydisc $\mathbb{G}_d$ and establish a realization theorem for functions in this class. We state and prove an interpolation theorem on $\mathbb{G}_d$ with interpolating functions belonging to the associated Schur-Agler type class. Moreover, Toeplitz corona and extension theorems are established for $\mathbb{G}_d$. We also extend the realization, interpolation, Toeplitz corona and extension theorems to a more general symmetrized family of domains $\Theta_d$.

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Works this paper leans on

51 extracted references · 19 canonical work pages

  1. [1]

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

    2007

  2. [2]

    M. B. Abrahamse,The Pick interpolation theorem for finitely connected domains, Michigan Math. J., 26 (1979), 195 – 203

    1979

  3. [3]

    Agler,On the representation of certain holomorphic functions defined on a polydisc, In: Topics in operator theory: Ernst D

    J . Agler,On the representation of certain holomorphic functions defined on a polydisc, In: Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., 48 (1990), 47 – 66

    1990

  4. [4]

    Agler, L

    J. Agler, L. Kosi ´nski and J. E. McCarthy,Complete norm-preserving extensions of holomorphic functions, Israel J. Math., 255 (2023), 251 – 263. FUNCTION THEORETIC ASPECTS OF THE SYMMETRIZED POLYDISC AND GENERALIZATION 25

    work page 2023

  5. [5]

    Agler, Z

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

    2015

  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, vii+108 pp

    work page 2019

  7. [7]

    Agler and J

    J. Agler and J. E. McCarthy,Nevanlinna-Pick interpolation on the bidisk, J. Reine Angew. Math., 506 (1999), 191 – 204

    1999

  8. [8]

    Agler and J

    J. Agler and J. E. McCarthy,Pick interpolation and Hilbert function spaces, Grad. Stud. Math., 44, Amer. Math. Soc., Providence, RI, 2002; MR1882259

    2002

Show all 51 references
  1. [9]

    Agler and J

    J. Agler and J. E. McCarthy,Norm preserving extensions of holomorphic functions from subvarieties of the bidisk, Ann. of Math., 157 (2003), 289 – 312

    2003

  2. [10]

    Agler and N

    J. Agler and N. J. Young,A commutant lifting theorem for a domain inC 2 and spectral interpolation, J. Funct. Anal., 161 (1999), 452 – 477

    1999

  3. [11]

    Agler and N

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

    2000

  4. [12]

    Agler and N

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

    2003

  5. [13]

    Agler and N

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

    2004

  6. [14]

    Agler and N

    J. Agler and N. J. Young,Realization of functions on the symmetrized bidisc, J. Math. Anal. Appl., 453 (2017), 227 – 240

    2017

  7. [15]

    Amar,On the Toeplitz corona problem, Publ

    E. Amar,On the Toeplitz corona problem, Publ. Mat., 47 (2003), 489 – 496

    2003

  8. [16]

    Ambrozie,Remarks on the operator-valued interpolation for multivariable bounded analytic functions, Indiana Univ

    C.-G. Ambrozie,Remarks on the operator-valued interpolation for multivariable bounded analytic functions, Indiana Univ. Math. J., 53 (2004), 1551 – 1576

    2004

  9. [17]

    And ˆo,On a pair of commutative contractions, Acta Sci

    T. And ˆo,On a pair of commutative contractions, Acta Sci. Math. (Szeged), 24 (1963), 88 – 90

    1963

  10. [18]

    A. V . Arkhangel’skii and L. S. Pontryagin,General topology-I, Springer, Berlin, 1990

    1990

  11. [19]

    Arveson,Interpolation problems in nest algebras, J

    W. Arveson,Interpolation problems in nest algebras, J. Funct. Anal., 20 (1975), 208 – 233

    1975

  12. [20]

    J. A. Ball and M. D. Guerra Huam ´an,Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting, Complex Anal. Oper. Theory, 7 (2013), 529 – 575

    2013

  13. [21]

    J. A. Ball and T. T. Trent,Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpo- lation in several variables, J. Funct. Anal., 157 (1998), 1 – 61

    1998

  14. [22]

    Biswas and V

    A. Biswas and V . S. Chandel,Interpolating sequences for the Banach algebras generated by a class of test functions, Oper. Matrices, 17 (2023), 295–315

    2023

  15. [23]

    Biswas, G

    S. Biswas, G. Ghosh, E. K. Narayanan and S. S. Roy,Contractive Hilbert modules on quotient domains, arxiv: 2409.11101

    arXiv

  16. [24]

    Bhattacharyya, S

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

    2012

  17. [25]

    Bhattacharyya and H

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

    2018

  18. [26]

    Bhattacharyya and H

    T. Bhattacharyya and H. Sau,Interpolating sequences and the Toeplitz-Corona theorem on the symmetrized bidisk, J. Operator Theory, 87 (2022), 435 – 459

    2022

  19. [27]

    Bhattacharyya, A

    T. Bhattacharyya, A. Biswas and V . S. Chandel,On the Nevanlinna problem–characterzation of all Schur-Agler class solutions affiliated with a given kernel, Studia Math., 255 (2020), 83 – 107

    2020

  20. [28]

    Carleson,Interpolations by bounded analytic functions and the corona problem, Ann

    L. Carleson,Interpolations by bounded analytic functions and the corona problem, Ann. of Math., 76 (1962), 547 – 559

    1962

  21. [29]

    Costara,The2×2spectral Nevanlinna-Pick problem, J

    C. Costara,The2×2spectral Nevanlinna-Pick problem, J. London Math. Soc., 71 (2005), 684 – 702

    2005

  22. [30]

    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

  23. [31]

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

    1981

  24. [32]

    M. A. Dritschel, S. Marcantognini and S. McCullough,Interpolation in semigroupoid algebras, J. Reine Angew. Math., 606 (2007), 1 – 40

    2007

  25. [33]

    M. A. Dritschel and S. McCullough,Test functions, kernels, realizations and interpolation, in: Operator theory, structured matrices, and dilations, Theta Ser. Adv. Math., 7 (2007), 153 – 179

    2007

  26. [34]

    Eschmeier and M

    J. Eschmeier and M. Putinar,Spherical contractions and interpolation problems on the unit ball, J. Reine Angew. Math., 542 (2002), 219 – 236

    2002

  27. [35]

    Foias and A

    C. Foias and A. E. Frazho,The commutant lifting approach to interpolation problem, Birkh ¨auser, Berlin, 1990. 26 SOURA V PAL AND NITIN TOMAR

    1990

  28. [36]

    B. A. Francis,A course in H ∞ control theory, Lecture Notes in Control and Information Sciences, V ol. 88, London: Springer-Verlag, 1987

    1987

  29. [37]

    S. Jain, S. Kumar, M. K. Mal and P. Pramanick,Function theory on tetrablock: realization, interpolation, exten- sion and Toeplitz Corona theorem, arXiv: 2505.23492

    arXiv

  30. [38]

    M. T. Jury, G. Knese and S. McCullough,Agler interpolation families of kernels, Oper. Matrices, 3 (2009), 571 – 587

    2009

  31. [39]

    D. K. Keshari, S. Nayak, A. Pal and B. Paul,RationalΘ n-inner function and its application in interpolation problem, arXiv: 2607.15662

    arXiv

  32. [40]

    Kosi ´nski and W

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

    2016

  33. [41]

    Kosi ´nski and W

    L. Kosi ´nski and W. Zwonek,Extension property and universal sets, Canad. J. Math., 73 (2021), 717 – 736

    2021

  34. [42]

    Mittal,Function theory on the quantum annulus and other domains, Thesis (Ph.D.)-University of Houston, ProQuest LLC, Ann Arbor, MI, 2010, 141 pp

    M. Mittal,Function theory on the quantum annulus and other domains, Thesis (Ph.D.)-University of Houston, ProQuest LLC, Ann Arbor, MI, 2010, 141 pp

    2010

  35. [43]

    Mujica,Complex Analysis in Banach Spaces, North-Holland Mathematics Studies 120, North-Holland, 1986

    J. Mujica,Complex Analysis in Banach Spaces, North-Holland Mathematics Studies 120, North-Holland, 1986

    1986

  36. [44]

    Nevanlinna, ¨Uber beschr¨ankte Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen, Ann

    R. Nevanlinna, ¨Uber beschr¨ankte Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen, Ann. Acad. Sci. Fenn. Ser. A, 13 (1919), 1 – 71

    1919

  37. [45]

    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

  38. [46]

    Pal and N

    S. Pal and N. Tomar,Operators associated with the pentablock and their relations with biball and symmetrized bidisc, Ann. Fenn. Math., 51 (2026), 287 – 324

    2026

  39. [47]

    Pal and N

    S. Pal and N. Tomar,Realization, interpolation and extension on the pentablock with applications toD 2,G 2, arXiv: 2606.00760

    arXiv

  40. [48]

    Pal and N

    S. Pal and N. Tomar,A Toeplitz corona theorem for the pentablock and applications, arXiv: 2606.00850

    arXiv

  41. [49]

    Pick, ¨Uber die Beschr¨ankungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math

    G. Pick, ¨Uber die Beschr¨ankungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann., 77 (1916), 7 – 23

    1916

  42. [50]

    Scheidemann,Introduction to complex analysis in several variables, Birkh¨auser Verlag, Basel, 2005

    V . Scheidemann,Introduction to complex analysis in several variables, Birkh¨auser Verlag, Basel, 2005

    2005

  43. [51]

    F. -H. Vasilescu,Analytic functional calculus and spectral decompositions, 1 (1982), Math. Appl. (East European Ser.), D. Reidel Publishing Co., Dordrecht. (Sourav Pal) MATHEMATICSDEPARTMENT, INDIANINSTITUTE OFTECHNOLOGYBOMBAY, POWAI, MUMBAI - 400076, INDIA. Email address:sour...

    1982

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