A new Banach space defined by absolute Jordan totient means
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초록
In the present study, we have constructed a new Banach series space $\left\vert \Upsilon ^{r}\right\vert _{p}^{u}$ by using concept of absolute Jordan totient summability $\left\vert \Upsilon ^{r},u_{n}\right\vert _{p}$ which is derived by the infinite regular matrix of the Jordan's totient function. Also, we prove that the series space $\left\vert \Upsilon ^{r}\right\vert _{p}^{u}$ is linearly isomorphic to the space of all $p$-absolutely summable sequences $\ell _{p}$ for $p\geq 1$. Moreover, we compute the $\alpha $-$,\beta $- and $\gamma $- duals of this space and construct Schauder basis for the series space $\left\vert \Upsilon^{r}\right\vert _{p}^{u}.$ Finally, we characterize the classes of infinite matrices $\left( \left\vert \Upsilon ^{r}\right\vert _{p}^{u},X\right) $ and $\left( X,\left\vert \Upsilon ^{r}\right\vert _{p}^{u}\right) ,$ where $X$ is any given classical sequence spaces $\ell _{\infty },$ $c,$ $c_{0}$ and $\ell _{1}$.
Abstract
In the present study, we have constructed a new Banach series space $\left\vert \Upsilon ^{r}\right\vert _{p}^{u}$ by using concept of absolute Jordan totient summability $\left\vert \Upsilon ^{r},u_{n}\right\vert _{p}$ which is derived by the infinite regular matrix of the Jordan's totient function. Also, we prove that the series space $\left\vert \Upsilon ^{r}\right\vert _{p}^{u}$ is linearly isomorphic to the space of all $p$-absolutely summable sequences $\ell _{p}$ for $p\geq 1$. Moreover, we compute the $\alpha $-$,\beta $- and $\gamma $- duals of this space and construct Schauder basis for the series space $\left\vert \Upsilon^{r}\right\vert _{p}^{u}.$ Finally, we characterize the classes of infinite matrices $\left( \left\vert \Upsilon ^{r}\right\vert _{p}^{u},X\right) $ and $\left( X,\left\vert \Upsilon ^{r}\right\vert _{p}^{u}\right) ,$ where $X$ is any given classical sequence spaces $\ell _{\infty },$ $c,$ $c_{0}$ and $\ell _{1}$.
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