Basic Classes of Linear Operators

Gohberg, Israel; Goldberg, Seymour; Kaashoek, M. A.

doi:10.1007/978-3-0348-7980-4

Cited by 185 publications

(166 citation statements)

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“…Some references for these results are Zhu (2007), Gohberg & Krejn (1971), Gohberg et al (1990), Kadison & Ringrose (1997a,b), Ringrose (1971).…”

Section: Background Resultsmentioning

confidence: 97%

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Tests for separability in nonparametric covariance operators of random surfaces

Tavakoli¹,

Pigoli²,

Aston³

2017

Contributions to Statistics

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Abstract. The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional data analysis settings, where this assumption is most relevant. We propose here to test separability by focusing on K-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full non-separable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.

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“…Some references for these results are Zhu (2007), Gohberg & Krejn (1971), Gohberg et al (1990), Kadison & Ringrose (1997a,b), Ringrose (1971).…”

Section: Background Resultsmentioning

confidence: 97%

“…Gohberg et al 1990, Kadison & Ringrose 1997a, Ringrose 1971. Let H be a real separable Hilbert space (that is, a Hilbert space with a countable orthonormal basis), whose inner product and norm are denoted by ·, · and · , respectively.…”

Section: Notationmentioning

confidence: 99%

Tests for separability in nonparametric covariance operators of random surfaces

Tavakoli¹,

Pigoli²,

Aston³

2017

Contributions to Statistics

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“…The following objects arose by independent reasons in spectral theory of non-selfadjoint operators, system theory, and representation theory of infinite-dimensional groups (see [12], [13], [29], [2], [3], [4], [7], [23], [24], [15]). …”

Section: Colligationsmentioning

confidence: 99%

Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument

Neretin

2017

Funct Anal Its Appl

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We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of size α + ∞ + · · ·+∞ (m times). Consider the subgroup K = U(∞), which consists of block diagonal unitary matrices with a block 1 of size α and a matrix u ∈ U(∞) repeated m times. It appears that there is a natural multiplication on the conjugacy classes G//K. We construct 'spectral data' of conjugacy classes, which visualize the multiplication and are sufficient for a reconstruction of a conjugacy class.MSC 22E66. 47A48, 15A72, 16W22

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“…We begin by noting that ω(f ) ∈ W 2×2 + . By the result for invertiblity of matrix-valued functions in the Wiener algebra W 2×2 + (see, e.g., [9]), the condition det(ω(f ))(z) = 0 for z ∈ B ∩ C implies that ω(f ) is invertible in W 2×2 + . Let G ∈ W 2×2 + be so that ω(f )G = I 2 on B ∩ C. Similar to the proof of Theorem 2.5 we get that…”

Section: If We Setmentioning

confidence: 99%