The following representation in terms of the spectral family is of interest in itself.

Lemma 2.1.

Let

be a self-adjoint operator in the Hilbert space

with the spectrum

for some real numbers

and let

be its spectral family. If

is a continuous function on

with

, then one has the representation

Proof.

which is an equality of interest in itself.

Since

are projections, we have

for any

and then we can write

Integrating by parts in the Riemann-Stieltjes integral and utilizing the spectral representation (1.1), we have

which together with (2.3) and (2.2) produce the desired result (2.1).

The following vector version may be stated as well.

Corollary 2.2.

With the assumptions of Lemma 2.1 one has the equality

for any
.

The following result that provides some bounds for continuous functions of bounded variation may be stated as well.

Theorem 2.3.

Let

be a self-adjoint operator in the Hilbert space

with the spectrum

for some real numbers

, and let

be its spectral family. If

is a continuous function of bounded variation on

with

, then we have the inequality

for any
.

Proof.

It is well known that if

is a bounded function,

is of bounded variation, and the Riemann-Stieltjes integral

exists, then the following inequality holds:

where
denotes the total variation of
on
.

Utilising this property and the representation (2.5), we have by the Schwarz inequality in Hilbert space

that

for any
.

Since

are projections, in this case we have

then from (2.8), we deduce the first part of (2.6).

Now, by the same property (2.7) for vector-valued functions

with values in Hilbert spaces, we also have

for any
and
.

Since
in the operator order, then
which gives that
, that is,
for any
, which implies that
for any
. Therefore,
which together with (2.10) prove the last part of (2.6).

The case of Lipschitzian functions is as follows.

Theorem 2.4.

Let

be a self-adjoint operator in the Hilbert space

with the spectrum

for some real numbers

, and let

be its spectral family. If

is a Lipschitzian function with the constant

on

and with

, then one has the inequality

for any
.

Proof.

Recall that if

is a Riemann integrable function and

is Lipschitzian with the constant

, that is,

then the Riemann-Stieltjes integral

exists and the following inequality holds:

Now, on applying this property of the Riemann-Stieltjes integral, then we have from the representation (2.5) that

for any
and the first inequality in (2.11) is proved.

for any
.

If we use the vector-valued version of the property (2.13), then we have

for any
and the second part of (2.11) is proved.

Further on, by applying the double-integral version of the Cauchy-Buniakowski-Schwarz inequality, we have

for any
.

Now, by utilizing the fact that

are projections for each

, then we have

for any
.

If we integrate by parts and use the spectral representation (1.2), then we get

and by (2.18), we then obtain the following equality of interest:

for any
.

On making use of (2.20) and (2.17), we then deduce the third part of (2.11).

Finally, by utilizing the elementary inequality in inner product spaces

for any
, which proves the last inequality in (2.11).

The case of nondecreasing monotonic functions is as follows.

Theorem 2.5.

Let

be a self-adjoint operator in the Hilbert space

with the spectrum

for some real numbers

, and let

be its spectral family. If

is a monotonic nondecreasing function on

, then one has the inequality

for any
.

Proof.

From the theory of Riemann-Stieltjes integral, it is also well known that if

is of bounded variation and

is continuous and monotonic nondecreasing, then the Riemann-Stieltjes integrals

and

exist and

Now, on applying this property of the Riemann-Stieltjes integral, we have from the representation (2.5) that

for any
, which proves the first inequality in (2.23).

On utilizing the Cauchy-Buniakowski-Schwarz-type inequality for the Riemann-Stieltjes integral of monotonic nondecreasing integrators, we have

for any
.

and, integrating by parts in the Riemann-Stieltjes integral, we have

for any
.

On making use of the equalities (2.28), we have

for any
.

Therefore, we obtain the following equality of interest in itself as well:

for any

On making use of the inequality (2.26), we deduce the second inequality in (2.23).

The last part follows by (2.21), and the details are omitted.