query: On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint.
positive_passages.docid: 19139
positive_passages.text: $T:\ell_2\to \ell_2, T(x_1,x_2,\dotsc)=(x_1,x_2/2,x_3/3,\dotsc)$
negative_passages.docid: 71011
negative_passages.text: Criteria of compactness of an operator
negative_passages.docid: 66953
negative_passages.text: How to prove Laplace transform is bounded on $L^2(\mathbb{R}_+)$?
negative_passages.docid: 52791
negative_passages.text: Invertibility of an operator $T$ if $\|I-T\|<1$
negative_passages.docid: 60685
negative_passages.text: The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $
negative_passages.docid: 71848
negative_passages.text: Show that $(L^{p},\|\|_{p})$ is a Banach space.
negative_passages.docid: 108781
negative_passages.text: $A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?
negative_passages.docid: 80618
negative_passages.text: Surjective bounded operator in Banach spaces without bounded right-inverse
negative_passages.docid: 151046
negative_passages.text: Closure of the invertible operators on a Banach space
negative_passages.docid: 65395
negative_passages.text: Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
negative_passages.docid: 116558
negative_passages.text: Consider two linear operators $\hat{A}$ and $\hat{B}$. Show that $e^{\hat{A}}e^{\hat{B}}=e^{\hat{A}+\hat{B}}e^{[\hat{A},\hat{B}]/2}$
negative_passages.docid: 87494
negative_passages.text: $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
negative_passages.docid: 125776
negative_passages.text: If space of bounded operators L(V,W) is Banach, V nonzero, then W is Banach (note direction of implication)
negative_passages.docid: 56102
negative_passages.text: Prove that there is no non-zero linear operator on $\mathbb C^{2}$ such that $(\alpha | T\alpha) = 0$
negative_passages.docid: 11558
negative_passages.text: How to show that the limit of compact operators in the operator norm topology is compact
negative_passages.docid: 8805
negative_passages.text: Bounded linear operators that commute with translation
negative_passages.docid: 80346
negative_passages.text: If every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal operator