Necessary and Sufficient Condition for Diagonal Operator to be Invertible

Theorem
Let $\mathbb F \in \set {\R, \C}$.

Let $\sequence {\lambda_n}_{n \mathop \in \N_{> 0} }$ be a bounded sequence in $\mathbb F$.

Let $\ell^2$ be the $2$-sequence space.

Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space on $\ell^2$.

Let $\Lambda \in \map {CL} {\ell^2}$ be the diagonal operator such that:


 * $\forall \mathbf x := \tuple {x_1, x_2, \ldots} \in \ell^2 : \map \Lambda {\mathbf x} = \tuple {\lambda_1 \cdot x_1, \lambda_2 \cdot x_2, \ldots}$

Then $\Lambda$ is invertible in $\map {CL} {\ell^2}$ $\ds \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} } > 0$ where $\inf$ denotes the infimum.

Necessary Condition
Suppose $\ds \inf_{n \mathop \in \N_{> 0} } \sequence {\size {\lambda_n} } > 0$.

Then:

Moreover:

Let $V : \ell^2 \to \ell^2$ be a mapping such that:


 * $\ds \forall \mathbf a := \sequence {a_n}_{n \mathop \in \N} = \tuple {a_1, a_2, \ldots} : \map V {\mathbf a} = \tuple {\frac {a_1} {\lambda_1}, \frac {a_2} {\lambda_2}, \ldots}$

Then $V \in \map {CL} {\ell^2}$.

Furthremore:

Therefore:


 * $\Lambda V = V \Lambda = I$

Hence, $\Lambda$ is invertible.

Finally, the unique inverse of $\Lambda$ is $\Lambda^{-1} = V$

Sufficient Condition
Suppose $\Lambda$ is invertible in $\map {CL} {\ell^2}$.

Then there is a unique inverse $\Lambda^{-1}$ of $\Lambda$:


 * $\ds \exists \Lambda^{-1} \in \map {CL} {\ell^2} : \Lambda \circ \Lambda^{-1} = \Lambda^{-1} \circ \Lambda = I$

We have that P-Sequence Space with P-Norm forms Normed Vector Space.

Hence:

Let $\mathbf x := \mathbf e_k = \tuple {\underbrace{0, \ldots, 0}_{k - 1}, 1, 0, \ldots}$

Then:

Therefore:


 * $\ds \inf_{k \mathop \in \N_{> 0} } \size {\lambda_k} \ge \frac 1 {\norm {\Lambda^{-1} } } > 0$