The paragraph is given in the following pictures:
If $A$ is an operator from $H_{1}$ to $H_{2}$,
The paragraph is divided into 5 sentences. I could not understand the first, second, forth and fifth sentences.
Could anyone explain them for me please?
Thanks!
Let $\mathfrak{F}(S)$ denotes the set of filters (including the improper filter) on a poset $S$, ordered reversely to set theoretic inclusion of filters. Let $Da$ for a lattice element $a$ denote its sublattice $\{ x \mid x \leq a \}$. Let $Z(X)$ denotes the set of complemented elements of the la...
Let $T:l_{2} \rightarrow l_{2}$ be defined by
$$T(x) = \left(x_{1},\frac{x_{2}}{2},\frac{x_{3}}{4},\ldots,\frac{x_{n}}{2^{n-1}},\ldots \right)$$
where $x = (x_{1},x_{2},.....).$
Find Im$T$.
I know that the kernel of this operator is $\{0\}$ and I know that the sum of the dim Im$T$ + dim ker$...
Let $n \in \mathbb{N^{*}}$ and define $T_{n}:l_{2} \rightarrow l_{2}$ by $$T_{n}(x) = (\frac{x_{n}}{n},\frac{x_{n + 1}}{n + 1},... ).$$
where $x = (x_{1},x_{2},.....).$
Then,$$T_{n}^{*}(x) = (0,...,0,\frac{x_{1}}{n},\frac{x_{2}}{n + 1},... ).$$
Where the zeros in the adjoint are $n-1$ times.
...
Let $n \in \mathbb{N^{*}}$ and define $T_{n}:l_{2} \rightarrow l_{2}$ by $$T_{n}(x) = (\frac{x_{n}}{n},\frac{x_{n + 1}}{n + 1},... ).$$
where $x = (x_{1},x_{2},.....).$
Then,$$T_{n}^{*}(x) = (0,...,0,\frac{x_{1}}{n},\frac{x_{2}}{n + 1},... ).$$
Where the zeros in the adjoint are $n-1$ times.
...
Um, the way the users "intuition" and "Idonotknow" ask their questions, I assume they are held by the same person. Their recent questions should be tagged "functional analysis" instead of the a little bit more advanced "operator theory".
Let $n \in \mathbb{N^{*}}$ and define $T_{n}:l_{2} \rightarrow l_{2}$ by $$T_{n}(x) = (\frac{x_{n}}{n},\frac{x_{n + 1}}{n + 1},... ).$$
where $x = (x_{1},x_{2},.....).$
Then,$$T_{n}^{*}(x) = (0,...,0,\frac{x_{1}}{n},\frac{x_{2}}{n + 1},... ).$$
Where the zeros in the adjoint are $n-1$ times.
...
