Normal Singular Integral Operators with Cauchy Kernel on L 2

Nakazi, Takahiko; Yamamoto, Takanori

doi:10.1007/s00020-013-2104-y

Cited by 31 publications

(11 citation statements)

References 3 publications

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“…The explicit expressions of general solution and the conditions of solvability are obtained in class 2 [− , ]. Therefore, this paper generalizes some results for [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

“…Various types of boundary value problems for analytic functions and singular integral equations have been deeply studied and widely applied to practical problem (see [1][2][3]). In the theory of integral equations, the convolution type integral equations and singular integral equations are two important classes of equations, which had been studied by many mathematical workers and there were already rather complete theoretical systems (see [4,5]). These theories have been widely used in practical applications, such as engineering mechanics, fracture mechanics, and elastic mechanics (see [6,7]).…”

Section: Introductionmentioning

confidence: 99%

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Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

2017

Mathematical Problems in Engineering

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We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class 2 [− , ]. Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.

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“…The explicit expressions of general solution and the conditions of solvability are obtained in class 2 [− , ]. Therefore, this paper generalizes some results for [1][2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

2017

Mathematical Problems in Engineering

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“…Recently, T. Nakazi and T. Yamomoto [17] proved that for any α, β ∈ L ∞ , the singular integral operator S α,β defined by S α,β f = α P f + β Q f for f ∈ L 2 which has the following matrices…”

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confidence: 99%

“…17 Let B(z) a j z be a finite Blaschke product. Suppose thatϕ = ϕ 1 + ϕ 1 + c 1 1 − ( n j=1 a j ) n j=1 a j − z 1 − a j z and ψ = ϕ 1 + ϕ 1 + c 2 1 − ( n j=1 a j )…”

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confidence: 99%

On the Dilation of Truncated Toeplitz Operators

Ko¹,

Lee

2015

Complex Anal. Oper. Theory

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An operator S u ϕ,ψ ∈ L(L 2 ) is called the dilation of a truncated Toeplitz operator if for two symbols ϕ, ψ ∈ L ∞ and an inner function u,where P u denotes the orthogonal projection of L 2 onto the model space K 2 u = H 2 u H 2 and Q u = I − P u . In this paper, we study properties of the dilation of truncated Toeplitz operators on L 2 . In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators.

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“…If α − β is a constant, then the following theorem gives the descriptions of symbols of normal (and hyponormal) operators αP + βQ. Brown and Halmos ( [2]) proved that the Toeplitz operator T α is normal if and only if α satisfies the condition (2) of the following corollary for some c ∈ C. In [12], normal singular integral operator αP + βQ is considered without the condition that α − β is a constant.…”

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confidence: 99%

Majorization of Singular Integral Operators with Cauchy Kernel on $L^2$

Yamamoto¹

2014

Ann. Funct. Anal.

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Let a, b, c and d be functions in L 2 = L 2 (T, dθ/2π), where T denotes the unit circle. Let P denote the set of all trigonometric polynomials. Suppose the singular integral operators A and B are defined by A = aP + bQ and B = cP + dQ on P, where P is an analytic projection and Q = I − P is a co-analytic projection. In this paper, we use the Helson-Szegő type set (HS)(r) to establish the condition of a, b, c and d satisfying Af 2 ≥ Bf 2 for all f in P. If a, b, c and d are bounded measurable functions, then A and B are bounded operators, and this is equivalent to that B is majorized by A on L 2 , i.e., A * A ≥ B * B on L 2 . Applications are then presented for the majorization of singular integral operators on weighted L 2 spaces, and for the normal singular integral operators aP + bQ on L 2 when a − b is a complex constant.

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Normal Singular Integral Operators with Cauchy Kernel on L 2

Cited by 31 publications

References 3 publications

Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

On the Dilation of Truncated Toeplitz Operators

Majorization of Singular Integral Operators with Cauchy Kernel on $L^2$

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