Math Problems Vault

Problem 50

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(a)

Give an example of an unbounded linear transformation that is bounded on a basis.

(b)

Is there such an example that annihilates a basis?

(c)

Is there such an example that is bounded on each basis?

(d)

Give examples of operators of arbitrarily large norms that are bounded by $1$ on a basis.

(e)

Could all the operators in such an example be normal?

Problem 51

Every weakly bounded set of bounded linear transformations is bounded.

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Remark.

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A set $\mathbf{Q}$ of linear transformations (from $\mathbf{H}$ into $\mathbf{K}$ ) is weakly bounded if for each $f$ in $\mathbf{H}$ and each $g$ in $\mathbf{K}$ there exists a positive constant $\alpha (f,g)$ such that $\left| \left\langle Af, g \right\rangle \right| \leq \alpha (f,g)$ for all $A$ in $\mathbf{Q}$ . The set $\mathbf{Q}$ is strongly bounded if for each $f$ in $\mathbf{H}$ there exists a positive constant $\beta (f)$ such that $\left\| Af\right\| \leq \beta (f)$ for all $A$ in $\mathbf{Q}$ . Every bounded set is strongly bounded and every strongly bounded set is weakly bounded. To say that $\mathbf{Q}$ is bounded means that there exists a constant $\gamma$ such that $\left\| A\right\| \leq \gamma$ for all $A$ in $\mathbf{Q}$ .

Problem 52

If $\mathbf{H}$ and $\mathbf{K}$ are Hilbert spaces, and if $A$ is a bounded linear transformation that maps $\mathbf{H}$ one-to-one onto $\mathbf{K}$ , then $A$ is invertible.

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Remark.

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A bounded linear transformation $A$ from a Hilbert space $\mathbf{H}$ to a Hilbert space $\mathbf{K}$ is invertible if there exists a bounded linear transformation $B$ (from $\mathbf{K}$ into $\mathbf{H}$ ) such that $AB = 1$ (the identity operator on $\mathbf{K}$ ) and $BA = 1$ (the identity operator on $\mathbf{H}$ ).

Problem 53

(a)(b)

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(a)

It is possible for a subspace $\mathbf{M}$ to have a diminishable complement $\mathbf{N}$ , in the sense that there exists another complement $\mathbf{N}_0$ , with $\mathbf{N}_0 \subset \mathbf{N}$ , $\mathbf{N}_0 \neq \mathbf{N}$ .

(b)

Can a complement $\mathbf{N}$ be infinitely diminishable, in the sense that there exists a complement $\mathbf{N}_0$ with $\mathbf{N}_0 \subset \mathbf{N}$ and $\dim (\mathbf{N} \cap \mathbf{N}_0^{\perp })$ infinite?

Remark.

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A complement of a subspace $\mathbf{M}$ in a Hilbert space $\mathbf{H}$ is a subspace $\mathbf{N}$ such that $\mathbf{M} \cap \mathbf{N} = 0$ and $\mathbf{M} \vee \mathbf{N} = \mathbf{H}$ .

Problem 54

There exists a linear manifold $\mathbf{G}$ dense in a Hilbert space $\mathbf{H}$ such that $\dim \mathbf{G} \neq \dim \mathbf{H}$ .

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Problem 55

Is every maximal orthonormal set in an inner-product space total?

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Remark.

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A subset of a Hilbert (or, more generally, inner-product) space is total if its span is the entire space. A total orthonormal set is always maximal, i.e., not properly contained in any larger orthonormal set.

Problem 56

If there exists a one-to-one bounded linear transformation from a Hilbert space $\mathbf{H}$ into a Hilbert space $\mathbf{K}$ , then $\dim \mathbf{H} \leq \dim \mathbf{K}$ . If the image of $\mathbf{H}$ is dense in $\mathbf{K}$ , then equality holds.

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Problem 57

If $P$ and $Q$ are projections such that $\left\| P-Q\right\| < 1$ , then $P$ and $Q$ have the same rank.

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Problem 58

A linear transformation from a Hilbert space into a Hilbert space is closed if and only if it is bounded.

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Remark.

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The graph of a linear transformation $A$ (not necessarily bounded) between inner product spaces $\mathbf{H}$ and $\mathbf{K}$ (not necessarily complete) is the set of all those ordered pairs $\left\langle f, g \right\rangle$ (elements of $\mathbf{H} \oplus \mathbf{K}$ ) for which $Af = g$ . A linear transformation is called closed if its graph is a closed set.

Problem 59

If $\operatorname {ran} A \subset \operatorname {ran} B$ , does it follow that $A$ is left divisible by $B$ ?

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Remark.

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An operator $A$ is "left divisible" by an operator $B$ if there exists an operator $X$ such that $A = BX$ .

Problem 60

(a)(b)

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(a)

Is a symmetric linear transformation on an inner-product space $\mathbf{H}$ necessarily bounded?

(b)

What if $\mathbf{H}$ is a Hilbert space?

Remark.

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A linear transformation $A$ (not necessarily bounded) on an inner-product space $\mathbf{H}$ (not necessarily complete) is called symmetric if $\left\langle Af, g \right\rangle = \left\langle f, Ag \right\rangle$ for all $f$ and $g$ in $\mathbf{H}$ . (This neutral term is used rather than "Hermitian" or "self-adjoint," because in the customary approach to Hermitian operators, $A = A^{*}$ , boundedness is an assumption necessary for the very formulation of the definition.)