• Kyungpook Mathematical Journal
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• v.26 no.1
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• pp.43-48
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• 1986

• East Asian mathematical journal
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• v.9 no.1
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• pp.7-14
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• 1993

• The Pure and Applied Mathematics
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• v.31 no.4
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• pp.453-476
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• 2024

The Riemann-Stieltjes integrals of continuous functions with respect to a function of bounded variation can be represented by a regular, Borel, complex measure. In this paper, we study the link between the Riemann-Stieltjes integral and measure theory using this representation. Specifically, we investigate the Riemann-Stieltjes integrability and its measurability. Furthermore, we derive a criterion for Riemann-Stieltjes integrability through a method different from known proofs. In particular, we calculate the upper and lower Riemann-Stieltjes integrals with respect to a monotone increasing function.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.375-415
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• 2019

We study spaces of continuous-function-valued functions that have the property that composition with evaluation functionals induce $weak^*$ to norm continuous maps to ${\ell}^p$ space ($p{\in}(1,\;{\infty})$). Versions of $H{\ddot{o}}lder^{\prime}s$ inequality and Riesz representation theorem are proved to hold on these spaces. We prove a version of Dixmier's theorem for spaces of function-valued matrix operators on these spaces, and an analogue of the trace formula for operators on Hilbert spaces. When the function space is taken to be the complex field, the spaces are just the ${\ell}^p$ spaces and the well-known classical theorems follow from our results.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.655-668
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• 2012

An operator $T\;{\varepsilon}\;B(\mathcal{H})$ is said to be $k$-quasi-paranormal operator if $||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$ for every $x\;{\epsilon}\;\mathcal{H}$, $k$ is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of $k$-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically $k$-quasi-paranormal operator has Bishop's property ($\beta$), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for $f(T)$ for every $f\;{\epsilon}\;H({\sigma}(T))$; (ii) generalized a - Browder's theorem holds for $f(S)$ for every $S\;{\prec}\;T$ and $f\;{\epsilon}\;H({\sigma}(S))$; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.