Абонент телефонує мобільному оператору.

Because the verb телефонувати takes the dative case for the person or organization being called.

• телефонувати кому? = to call / phone whom?
• So оператор becomes оператору in the dative singular.

This is different from English, where we usually just say call someone without changing the noun form.

Because adjectives must agree with the noun they describe in case, number, and gender.

Here:

• оператору = masculine, singular, dative
• so мобільний also has to be masculine, singular, dative
• therefore: мобільному оператору

So the phrase changes as a unit:

• мобільний оператор = nominative
• мобільному оператору = dative

Телефонує is the 3rd person singular present tense form of телефонувати.

That means it matches a singular subject:

• абонент телефонує = the subscriber is calling / calls

It is an imperfective verb, so it describes an action in progress, a repeated action, or a general present action depending on context.

It can mean either one, depending on context.

Ukrainian present tense often covers both:

• is calling
• calls

So Абонент телефонує мобільному оператору could mean:

• The subscriber is calling the mobile operator
• The subscriber calls the mobile operator

If you want to make the ongoing meaning clearer, context words like зараз (now) help:

• Абонент зараз телефонує мобільному оператору.

Because Ukrainian often uses case endings instead of a preposition.

In English:

• call the operator
• or historically phone to someone is not used, but the relationship is still expressed syntactically

In Ukrainian, with телефонувати, the dative ending already shows the idea of to someone:

• оператору = to the operator

So no extra preposition is needed.

Абонент is a fairly common formal word meaning something like:

• subscriber
• customer
• user of a phone/network/service

In telecom contexts, it is very natural. In everyday speech, depending on context, people might also say клієнт (client/customer) instead.

Grammatically, абонент is a masculine singular noun, but it can be used generically when the person’s gender is not specified.

Usually мобільний оператор means the mobile carrier / service provider.

So in this sentence, the idea is most likely that the subscriber is calling their mobile network provider.

However, the word оператор by itself can also mean a human operator / agent, so context matters. In telecom language, though, мобільний оператор normally refers to the company.

Yes. Ukrainian word order is more flexible than English because the case endings show each word’s role.

The neutral order here is:

• Абонент телефонує мобільному оператору.

But you could also say:

• Мобільному оператору телефонує абонент.

That still means the same basic thing, but the emphasis changes. The second version puts more focus on мобільному оператору.

So word order in Ukrainian often affects emphasis, not the core grammar.

Because Ukrainian has no articles.

So:

• абонент can mean a subscriber or the subscriber
• мобільному оператору can mean to a mobile operator or to the mobile operator

Context tells you which one is meant.

That is why one Ukrainian sentence can correspond to more than one natural English translation.

The usual perfective partner is зателефонувати.

Compare:

• телефонувати = imperfective, ongoing / repeated / general
• зателефонувати = perfective, one completed act of calling

Examples:

• Абонент телефонує мобільному оператору.
  = The subscriber is calling / calls the mobile operator.

• Абонент зателефонував мобільному оператору.
  = The subscriber called the mobile operator.

So if you want to emphasize a single completed call, зателефонувати is often the better choice.