Firstly: 

Allaah has commanded the worshipper to beautify and adorn himself for prayer, as He says (interpretation of the meaning): 

“O Children of Adam! Take your adornment (by wearing your clean clothes) while praying”

[al-A’raaf 7:31]

Adorning oneself for prayer is something that is additional to covering the ‘awrah. Hence women are commanded to cover their heads when praying, although they are allowed to uncover their heads in front of their mahrams. 

Shaykh al-Islam Ibn Taymiyah said: 

With regard to adorning oneself for prayer, it is something that is additional to covering the ‘awrah, and it is based on the Qur’aan, the Sunnah and scholarly consensus. In the Qur’aan, there is the verse (interpretation of the meaning): “O Children of Adam! Take your adornment (by wearing your clean clothes) while praying” [al-A’raaf 7:31], which Allaah revealed because the mushrikeen used to circumambulate the Ka’bah naked. Every place of prostration is a mosque (masjid), and this indicates that covering oneself for prayer and tawaaf is what is meant by adorning oneself to worship Allaah. Hence the word adornment is used rather than the word covering, to demonstrate that what is meant is that a person should adorn himself and not limit it to simply covering. 

In the Sunnah, the Prophet (peace and blessings of Allaah be upon him) said: “Allaah does not accept the prayer of any woman who menstruates except with a head covering.” 

With regard to scholarly consensus, Abu Bakr ibn al-Mundhir said: The scholars are unanimously agreed that a free, adult woman must cover her head when she prays, and that if she prays with her entire head uncovered, then she must repeat the prayer. Others also narrated that there was consensus that covering is an essential condition. End quote. 

Sharh al-‘Umdah, 4/258, 259. 

Secondly: 

Once this is understood, although the worshipper must adorn himself for prayer, this adornment may vary from one country to another, according to their customs, including what is mentioned in the question, which is praying in a garment with half-sleeves or praying bare-headed. If the custom of that land is that the best adornment is to cover the head or to pray in a garment with full sleeves, then praying when one is dressed as mentioned in the question is contrary to what Allaah has enjoined, even if the prayer itself is valid. But if the custom of the people in that country is to dress as described in the question, then there is nothing wrong with praying dressed in this manner. 

It should be noted that the ruling also varies according to variations in the dress itself. The garment may have short sleeves and that may be customarily worn in some countries where it may be worn as an adornment, or a man may go to work in it and so on. In that case there is nothing wrong with praying in it. Or it may be the custom that this is not a kind of adornment, and it is something that a man wears at home only, or when sleeping only, in which case it is not appropriate to pray in it. 

There follow the fatwas of the scholars, which include the principle mentioned above, which is that this ruling depends on the customs in each country. 

1 – Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) was asked: an imam leads the people in prayer and he is not wearing anything on his head. What is the ruling on that? 

He replied: 

There is nothing wrong with that, because the head is not part of the ‘awrah, rather what is required is to pray in a lower and upper garment, because the Prophet (peace and blessings of Allaah be upon him) said: “No one of you should pray in a single garment with no part of it over his shoulders.” But if he wears his adornment and dresses fully, that is preferable, because Allaah says (interpretation of the meaning): 

“O Children of Adam! Take your adornment (by wearing your clean clothes) while praying”

[al-A’raaf 7:31]

But if he is in a country where it is not customary to cover the head, then there is nothing wrong with leaving it bare. End  quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz, 10/405, 406 

2 – He was also asked: Is it permissible to pray without a turban? Is it permissible for the imam who is leading the people in prayer to pray without a head cover? Is the taaqiyah (kufi, skull-cap) sufficient? 

He replied: 

There is nothing wrong with praying without a turban because the head is not ‘awrah, and it is not obligatory to cover it whilst praying, whether one is praying as an imam, praying alone or following an imam. But if wearing a turban is customary then it is better to do it, especially when praying with other people, because Allaah says (interpretation of the meaning): 

“O Children of Adam! Take your adornment (by wearing your clean clothes) while praying”

[al-A’raaf 7:31]

and this is part of adornment. 

It is well known that men who are in ihraam pray bareheaded, because they are not allowed to cover their heads whilst in ihraam. Hence it is known that there is nothing wrong with praying bareheaded. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz, 10/406 

3 – Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

It was narrated that Ibn ‘Umar said to his freed slave Naafi’: “Do you go and meet people bareheaded?” He said: “No.” He said: “Allaah has more right that you should be modest before Him.” This indicates that it is better to cover the head, but if we apply the words of Allaah - (interpretation of the meaning): “O Children of Adam! Take your adornment (by wearing your clean clothes) while praying” [al-A’raaf 7:31] – to this issue, we will see that covering the head is better among people who regard covering the head as a kind of adornment. But if we are among people who do not regard that as a kind of adornment, we do not say that covering it is better or that leaving it bare is better. It is proven that the Prophet (peace and blessings of Allaah be upon him) used to pray in a turban, and the turban is a head covering. End quote. 

Al-Sharh al-Mumti’, 2/166 

And Allaah knows best.