Project MUSE®: Mathematical Proceedings of the Royal Irish Academy - Latest Articles
https://muse.jhu.edu/journal/814
Project MUSE®: Latest articles in Mathematical Proceedings of the Royal Irish Academy.daily12026-05-13T00:00:00-05:00text/htmlen-USVol. 102A (2002) through current issueLatest Articles: Mathematical Proceedings of the Royal Irish AcademyTWOProject MUSE®Mathematical Proceedings of the Royal Irish Academy2009-00211393-7197Latest articles in Mathematical Proceedings of the Royal Irish Academy. Feed provided by Project MUSE®An RKHS approach to the indefinite Schwarz–Pick inequality on the bidisk
https://muse.jhu.edu/article/959701
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Let
denote the open unit disk in the complex plane
. For any two points (z1, z2) and (w1, w2) in
2, we defineThis is a distance on
2 invariant under the automorphism group of
2. Seto [9, theorem 4.1] proved the following Schwarz–Pick type inequality for ρ:Theorem 1.1. If F :
is a holomorphic map on
2, then F satisfiesfor all (z1, z2), (w1, w2) ∈
2.He used the theory of Hilbert modules in the Hardy space over the bidisk to prove this result. In this note, we will give a generalisation of Theorem 1.1 as an application of the geometry of reproducing kernel Hilbert spaces (RKHS for short) with ‘the two-point Nevanlinna–Pick property’ (see Definition 2.3). This property means that an RKHS enjoys a two-point
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Project MUSE®https://muse.jhu.edu/2026-05-13T00:00:00-05:00https://muse.jhu.edu/journal/814/image/coversmallAn RKHS approach to the indefinite Schwarz–Pick inequality on the bidisk2025-05-01text/htmlen-USAn RKHS approach to the indefinite Schwarz–Pick inequality on the bidisk2025-05-012025TWOProject MUSE®971522026-05-13T00:00:00-05:002025-05-01The Three Obdurate Conjectures of Differential Geometry
https://muse.jhu.edu/article/968272
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In a 1980s UC Berkeley lecture [21], Robert Osserman identified what he referred to as the three obdurate conjectures of differential geometry, all of which were open at that time: the Lawson Conjecture [22], which was proven in 2014 by Marques and Neves [27]; the Willmore Conjecture [38], which was proven in 2013 by Brendle [6]; and the Carathéodory Conjecture, whose proof has been published by the authors in [14] (see [15] [16] [17]).The three Conjectures were:Carathéodory Conjecture (1924): Every strictly convex surface in ℝ3 contains at least two umbilic points (points at which the principal curvatures are equal).Willmore Conjecture (1969): The Willmore energy of an embedded torus T in satisfieswhere H is the
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Project MUSE®https://muse.jhu.edu/2026-05-13T00:00:00-05:00https://muse.jhu.edu/journal/814/image/coversmallThe Three Obdurate Conjectures of Differential Geometry2025-08-26text/htmlen-USThe Three Obdurate Conjectures of Differential Geometry2025-08-262025TWOProject MUSE®725012026-05-13T00:00:00-05:002025-08-26Algebraic Operators on Banach Spaces
https://muse.jhu.edu/article/977728
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Throughout this note, we use X to denote an infinite-dimensional complex Banach space and L(X) to denote the Banach algebra of all bounded linear operators. If T ∈ L(X), by α(T) and β(T) we denote the dimension of the kernel ker T and the codimension of the range R(T) := T(X), respectively. An operator T ∈ L(X) is said to be upper semi-Fredholm, T ∈ Φ+(X), if α(T) < ∞ and T(X) is closed; T ∈ L(X) is said to be lower semi-Fredholm, T ∈ Φ−(X), if β(T) < ∞. T ∈ L(X) is said to be Fredholm if T ∈ Φ+(X) ∩ Φ−(X). It is well known that for a complex infinite -dimensional Banach space X the Fredholm region of an operator T ∈ L(X), defined bycannot coincide with ℂ.Recall that the ascent p := p(T) of an operator T is the
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Project MUSE®https://muse.jhu.edu/2026-05-13T00:00:00-05:00https://muse.jhu.edu/journal/814/image/coversmallAlgebraic Operators on Banach Spaces2025-12-16text/htmlen-USAlgebraic Operators on Banach Spaces2025-12-162025TWOProject MUSE®898862026-05-13T00:00:00-05:002025-12-16