6.7 KiB
Linear operators
Definition 1: a linear operator
T
is a linear mapping such that
- the domain
\mathscr{D}(T)
ofT
is a vector space and the range\mathscr{R}(T)
ofT
is contained in a vector space over the same field as\mathscr{D}(T)
.\forall x, y \in \mathscr{D}(T): T(x + y) = Tx + Ty
.\forall x \in \mathscr{D}(T), \alpha \in F: T(\alpha x) = \alpha Tx
.
Observe the notation; we Tx
and T(x)
are equivalent, most of the time.
Definition 2: let
\mathscr{N}(T)
be the null space ofT
defined as
\mathscr{N}(T) = {x \in \mathscr{D}(T) ;|; Tx = 0}.
We have the following properties.
Proposition 1: let
T
be a linear operator, then
\mathscr{R}(T)
is a vector space,\mathscr{N}(T)
is a vector space,- if
\dim \mathscr{D}(T) = n \in \mathbb{N}
then\dim \mathscr{R}(T) \leq n
.
??? note "Proof:"
Will be added later.
An immediate consequence of statement 3 is that linear operators preserve linear dependence.
Proposition 2: let
Y
be a vector space, a linear operatorT: \mathscr{D}(T) \to Y
is injective if
\forall x_1, x_2 \in \mathscr{D}(T): Tx_1 = Tx_2 \implies x_1 = x_2.
??? note "Proof:"
Will be added later.
Injectivity of T
is equivalent to \mathscr{N}(T) = \{0\}
.
??? note "Proof:"
Will be added later.
Theorem 1: if a linear operator
T: \mathscr{D}(T) \to \mathscr{R}(T)
is injective there exists a mappingT^{-1}: \mathscr{R}(T) \to \mathscr{D}(T)
such that
y = Tx \iff T^{-1} y = x,
for all
x \in \mathscr{D}(T)
, denoted as the inverse operator.
??? note "Proof:"
Will be added later.
Proposition 3: let
T: \mathscr{D}(T) \to \mathscr{R}(T)
be an injective linear operator, if\mathscr{D}(T)
is finite-dimensional, then
\dim \mathscr{D}(T) = \dim \mathscr{R}(T).
??? note "Proof:"
Will be added later.
Lemma 1: let
X,Y
andZ
be vector spaces and letT: X \to Y
andS: Y \to Z
be injective linear operators, then(ST)^{-1}: Z \to X
exists and
(ST)^{-1} = T^{-1} S^{-1}.
??? note "Proof:"
Will be added later.
We finish this subsection with a definition of the space of linear operators.
Definition 3: let
\mathscr{L}(X,Y)
denote the set of linear operators mapping from a vector spaceX
to a vector spaceY
.
From this definition the following theorem follows.
Theorem 2: let
X
andY
be vectors spaces, the set of linear operators\mathscr{L}(X,Y)
is a vector space.
??? note "Proof:"
Will be added later.
Therefore, we may also call \mathscr{L}(X,Y)
the space of linear operators.
Bounded linear operators
Definition 4: let
(X, \|\cdot\|_X)
and(Y,\|\cdot\|_Y)
be normed spaces over a fieldF
and letT: \mathscr{D}(T) \to Y
be a linear operator with\mathscr{D}(T) \subset X
. ThenT
is a bounded linear operator if
\exists c \in F \forall x \in \mathscr{D}(T): |Tx|_Y \leq c |x|_X.
In this case we may also define the set of all bounded linear operators.
Definition 5: let
\mathscr{B}(X,Y)
denote the set of bounded linear operators mapping from a vector spaceX
to a vector spaceY
.
We have the following theorem.
Theorem 3: let
X
andY
be vectors spaces, the set of bounded linear operators\mathscr{B}(X,Y)
is a subspace of\mathscr{L}(X,Y)
.
??? note "Proof:"
Will be added later.
Likewise, we may call \mathscr{B}(X,Y)
the space of bounded linear operators.
The smallest possible c
such that the statement in definition 4 still holds is denoted as the norm of T
in the following definition.
Definition 5: the norm of a bounded linear operator
T \in \mathscr{B}(X,Y)
is defined by
|T|{\mathscr{B}} = \sup{x \in \mathscr{D}(T) \backslash {0}} \frac{|Tx|_Y}{|x|_X},
with
X
andY
vector spaces.
The operator norm makes \mathscr{B}
into a normed space.
Lemma 2: let
X
andY
be normed spaces, the norm of a bounded linear operatorT \in \mathscr{B}(X,Y)
may be given by
|T|\mathscr{B} = \sup{\substack{x \in \mathscr{D}(T) \ |x|_X = 1}} |Tx|_Y,
and the norm of a bounded linear operator is a norm.
??? note "Proof:"
Will be added later.
Note that the second statement in lemma 2 is non trivial, as the norm of a bounded linear operator is only introduced by a definition.
Proposition 4: if
(X, \|\cdot\|)
is a finite-dimensional normed space, then every linear operator onX
is bounded.
??? note "Proof:"
Will be added later.
By linearity of the linear operators we have the following.
Theorem 4: let
X
andY
be normed spaces and letT: \mathscr{D}(T) \to Y
be a linear operator with\mathscr{D}(T) \subset X
. Then the following statements are equivalent
T
is bounded,T
is continuous in\mathscr{D}(T)
,T
is continuous in a point in\mathscr{D}(T)
.
??? note "Proof:"
Will be added later.
Corollary 1: let
T \in \mathscr{B}
and let(x_n)_{n \in \mathbb{N}}
be a sequence in\mathscr{D}(T)
, then we have that
x_n \to x \in \mathscr{D}(T) \implies Tx_n \to Tx
asn \to \infty
,\mathscr{N}(T)
is closed.
??? note "Proof:"
Will be added later.
Furthermore, bounded linear operators have the property that
|T_1 T_2| \leq |T_1| |T_2|,
for T_1, T_2 \in \mathscr{B}
.
??? note "Proof:"
Will be added later.
Theorem 5: if
X
is a normed space andY
is a Banach space, then\mathscr{B}(X,Y)
is a Banach space.
??? note "Proof:"
Will be added later.
Definition 6: let
T_1, T_2 \in \mathscr{L}
be linear operators,T_1
andT_2
are equal if and only if
\mathscr{D}(T_1) = \mathscr{D}(T_2)
,\forall x \in \mathscr{D}(T_1) : T_1x = T_2x
.
Restriction and extension
Definition 7: the restriction of a linear operator
T \in \mathscr{L}
to a subspaceA \subset \mathscr{D}(T)
, denoted byT|_A: A \to \mathscr{R}(T)
is defined by
T|_A x = Tx,
for all
x \in A
.
Furthermore.
Definition 8: the extension of a linear operator
T \in \mathscr{L}
to a vector spaceM
is an operator denoted by\tilde T: M \to \mathscr{R}(T)
such that
\tilde T|_{\mathscr{D}(T)} = T.
Which implies that \tilde T x = Tx\; \forall x \in \mathscr{D}(T)
. Hence, T
is the resriction of \tilde T
.
Theorem 6: let
X
be a normed space and letY
be Banach space. LetT \in \mathscr{B}(M,Y)
withA \subset X
, then there exists an extension\tilde T: \overline M \to Y
, with\tilde T
a bounded linear operator and\| \tilde T \| = \|T\|
.
??? note "Proof:"
Will be added later.