Let ℋ be a Hilbert space. An operator T in ℬ(ℋ) is said to be a contraction if T ≤1. (a) Show th

Let ℋ be a Hilbert space. An operator T in ℬ(ℋ) is said to...

Key Concepts

Invertible Operator

An invertible operator is a bounded linear operator that has a well-defined inverse, meaning every vector in the Hilbert space can be uniquely recovered by applying the inverse operator. Invertibility is crucial in many areas of analysis, including the study of operator equations and transformations, as it allows one to 'undo' the effect of an operator.

Operator Inequality

Operator inequalities involve comparing operators via the concept of positivity. For example, saying that one operator is less than or equal to another (A ? B) means that the difference B - A is a positive operator. This framework is essential for establishing the equivalence between contraction properties and conditions expressed in terms of operator differences.

Positive Operator

A positive operator on a Hilbert space is one for which the inner product ?Ax, x? is nonnegative for all vectors in the space. This notion of positivity allows the establishment of order relationships between operators and is critical when expressing conditions such as I - T*T being positive to characterize contractions.

Adjoint Operator

The adjoint of an operator is another operator that relates to the original via the inner product, satisfying the relationship ?Tx, y? = ?x, T*y? for all vectors in the space. It is fundamental for defining self-adjoint, unitary, and positive operators, and is key in characterizing contractions through conditions like I - T*T being positive.

Contraction

A contraction is a specific type of bounded linear operator whose operator norm is less than or equal to one. This property implies that the operator does not increase the length of any vector, playing a significant role in fixed-point theorems and stability analyses within Hilbert spaces.

Bounded Linear Operator

A bounded linear operator on a Hilbert space is a linear transformation that maps elements of the space in a way that the norm of the output is consistently controlled by the norm of the input. Boundedness ensures continuity and that the operator behaves well under limits, which is crucial for many analytic techniques in Hilbert space theory.

Hilbert Space

A Hilbert space is a complete inner product space that generalizes the notion of Euclidean space to infinite dimensions. It provides a framework for discussing geometric concepts like orthogonality, projection, and distance, which are essential to the study of linear operators and functional analysis.

Operator Norm

The operator norm is a measure of the 'size' or 'magnitude' of a bounded linear operator, defined as the supremum of the norms of the outputs over all unit input vectors. This norm is used to quantify the contraction property of an operator, with a contraction having a norm not exceeding one.

Transcript

00:01 All right, so in this question we're given a bounded linear operator between two norm spaces x and y.

00:06 We're told it is onto y, so it is surjective, and it satisfies this estimate of the form.

00:11 So there exists a b such that this holds for every x, and then we want to prove that then this implies that t is invertible and the inverse is bounded.

00:21 So first step, let's show that the inverse exists.

00:25 So you know an operator is invertible if it is injective and surjective, so if it is one -to -one and onto, we already have onto, so we only have to prove one -to -one.

00:36 So first step, the inverse exists, so we need to show that t is one -to -one or injective.

00:45 So you know how this goes.

00:47 Let's suppose x in x is such that t of x is equal to zero, and we want to show that x must be zero.

01:01 Then let's see what we can do.

01:03 I mean, we only have one assumption on x, so we must have to use it.

01:07 So i want to show that x is zero.

01:09 I will do so by showing that the norm of x is equal to zero.

01:13 So let's take the norm of x.

01:15 So we can use this estimate in here, in red.

01:20 We can use this estimate in here to write that norm of x is less than or equal to one over b times the norm of t of x.

01:31 This is valid for every x.

01:33 And now we're done, right? because by assumption, t of x is a zero vector.

01:37 The zero vector has norm equal to zero.

01:40 So this is equal to one over b times zero, which is zero.

01:45 So now the norm of x, this is always positive, right? non -negative.

01:48 It is above zero, but also below zero.

01:52 So we conclude that the norm of x is equal to zero, and then this implies that x is zero itself.

01:58 So we show that if a number is such that its image is zero, that number is zero...

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